Archive for the ‘Giới thiệu’ Category

Journal of Integer Sequences

Monday, October 15th, 2012

Current Volumes

 

Bài tập hình học

Friday, October 12th, 2012

Pictorial Geometry Index

  1. 1 + 27 = 12 + 16 Sangaku
  2. 120° Breeds 90° [Java]
  3. 3 Roads, 3 Travelers [Java]
  4. 3 Utilities Puzzle
  5. 3D Quadrilateral - a Coffin Problem
  6. 3-4-5 Triangle by a Kid
  7. 4 Travelers problem
  8. 6 to 9 Point Circle [Java]
  9. 60o Breeds 90o [Java]
  10. 7 = 2 + 5 Sangaku
  11. 9-point Circle as a Locus of Concurrency [Java]
  12. A better solution to a difficult sangaku problem
  13. A Case of Similarity [Java]
  14. A Chain of Touching Circles in a Polygon (a la Bui Quang Tuan) [Java]
  15. A Chain of Touching Circles in a Polygon (à la Evelyn) [Java]
  16. A Characterization of the Euler Line [Java]
  17. A Circle in an Isosceles triangle from the 2006 Itish MO
  18. A Circle Related to Incenter and Circumcenter [Java]
  19. A Circle Rolling in an Equilateral Triangle [Java]
  20. A Circle With Two Centers [Java]
  21. A Corrected Version of the Calculus Proof of the Pythagorean Theorem
  22. A Generalized Cavalieri-Zu Principle
  23. A Geometric Limit
  24. A Geometry Problem from 2012 IMO - Problem 5
  25. A Hard but Important Sangaku
  26. A Hinged Realization of a Plane Tessellation [Java]
  27. A Lemma of Equal Areas [Java]
  28. A Line in a Square Grid [Java]
  29. A Line in Triangle Through the Circumcenter [Java]
  30. A Multiplicative Identity of Areas in a Triangle [Java]
  31. A Note on Cavalieri’s Indivisibles
  32. A Parallelogram in Triangle [Java]
  33. A Possibly First Proof of the Concurrence of Altitudes
  34. A Problem In an Isosceles Right Triangle [Java]
  35. A Problem on an Icy Cone
  36. A problem with equilateral triangles [Java]
  37. A Proof Perigal and All Others After Him Missed
  38. A Property of Equiangular Polygons [Java]
  39. A Property of Rhombi [Java]
  40. A Property of the Line IO [Java]
  41. A Property of the Line IO: A Bundled Statement and Proof [Java]
  42. A Property of the Line IO: a proof from the Book [Java]
  43. A Relation in Triangle
  44. A Restored Sangaku
  45. A Sangaku Follow-Up on an Archimedes’ Lemma [Java]
  46. A Sangaku: Two Unrelated Circles [Java]
  47. A Sangaku with an Egyptian Attachment [Java]
  48. A Sangaku with Many Circles and Some [Java]
  49. Semi-regular Tessellation on Hinges A [Java]
  50. Semi-regular Tessellation on Hinges B [Java]
  51. A Stronger Triangle Inequality [Java]
  52. A Sushi Morsel
  53. A Three Pegs Question [Java]
  54. A Triangle of Antreas Hatzipolakis [Java]
  55. A Trigonometric Identity in a Right Triangle from the 1960 IMO
  56. A Trigonometric Solution to a Difficult Sangaku Problem
  57. An Area Inequality
  58. About a Line and a Triangle [Java]
  59. Abul Wafa al-Buzjani’s Proof of the Pythagorean Theorem [Java]
  60. Acute Angle Bisectors in a Right Triangle [Java]
  61. Adams’ Circle [Java]
  62. Addition and Subtraction Formulas for Sine and Cosine
  63. Addition and Subtraction Formulas for Sine and Cosine II
  64. Algebraic proof of the theorem of butterflies in quadrilaterals
  65. All Triangles Are Isosceles [Java]
  66. Altitudes [Java]
  67. Altitudes and the Euler Line [Java]
  68. Altitudes and the Power of a Point [Java]
  69. Altitudes Concur at a Point: a Second Look
  70. An Extension of van Schooten’s Theorem
  71. An Identity in a Regular Heptagon
  72. An Isoperimetric Theorem [Java]
  73. An Old Japanese Theorem
  74. Angle: An Illustrated Classification [Java]
  75. Angle Bisector in Equilateral Trapezoid
  76. Angle Bisector in Parallelogram [Java]
  77. Angle Bisectors In Rectangle [Java]
  78. Angle Bisector in Touching Circles [Java]
  79. Angle Bisectors [Java]
  80. Angle Bisectors and Perpendiculars in a Quadrilateral [Java]
  81. Angle Bisectors in Ellipse [Java]
  82. Angle Bisectors in Ellipse II [Java]
  83. Angle Bisectors in a Quadrilateral [Java]
  84. Angle Bisectors in a Quadrilateral - Cyclic and Otherwise [Java]
  85. Angle Bisectors On Circumcircle [Java]
  86. Angle Subtended by a Diameter [Java]
  87. Angle Preservation Property [Java]
  88. Angle Trisection, the Impossibility of Thereof
  89. Angle Trisection by Archimedes [Java]
  90. Angle Trisection by Hippocrates [Java]
  91. Angle Trisection by Paul Vjecsner [Java]
  92. Angle Trisectors on Circumcircle [Java]
  93. Angle Trisection by Paper Folding
  94. Angles in a Cube I [Java]
  95. Angles in a Cube II [Java]
  96. Angles in a Cube III [Java]
  97. Angles in Triangle Add to 180o
  98. Angles Inscribed in an Absent Circle [Java]
  99. Another Geometry Problem from 2012 IMO - Problem 1
  100. Anticomplementary Triangle Surprise [Java]
  101. Antiparallel [Java]
  102. Antiparallel via Three Reflections [Java]
  103. APMO 1995 Problem 3
  104. APMO 1998, Problem 4
  105. APMO 2002, Problem 3
  106. APMO 2003, Problem 2
  107. APMO 2004, Problem 2
  108. Apollonian Circles Theorem [Java]
  109. Apollonian Gasket [Java]
  110. Apollonius Problem [Java]
  111. Approximate Construction of Regular Pentagon by A. Dürer
  112. Arbelos — the Shoemaker’s Knife [Java]
  113. Archimedes’ Book of Lemmas
  114. Archimedes’ Law of the Lever [Java]
  115. Archimedes’ Method
  116. Archimedes’ Twins in the Edo Period
  117. Area of a Bicentric Quadrilateral
  118. Area of a Circle by Leonardo da Vinci [Java]
  119. Area of a Circle by Rabbi Abraham bar Hiyya Hanasi [Java]
  120. Area of Median Triangle [Java]
  121. Area of Parallelogram [Java]
  122. Area of Parallelogram Formula by Shearing
  123. Area of the Union of Two Squares [Java]
  124. Area of Triangle [Java]
  125. Areas and Centroid in a Triangle [Java]
  126. Areas In Circle
  127. Equal Areas in Regular 2n-gons [Java]
  128. Areas in Square by Dissection [Java]
  129. Arithmetic-Geometric Mean Inequality
  130. Arithmetic Mean Sangaku [Java]
  131. Arithmetic Mean Sangaku II [Java]
  132. Around the Incircle [Java]
  133. Around the Incircle in a Polygon [Java]
  134. Artisans vs Mathematicians
  135. Asian Pacific Kite [Java]
  136. Assigning Numbers to Points in the Plane
  137. Assimilation Illusion [Java]
  138. Asymmetric Propeller [Java]
  139. Asymmetric Propeller by Plane Tiling [Java]
  140. Axis of Similitude of Three Circles
  141. Barbier, The Theorem of [Java]
  142. Barycentric coordinates
  143. Bender: A Visual Illusion [Java]
  144. Between Major and Minor Circles [Java]
  145. Bevan’s Point and Theorem [Java]
  146. Bimedians and Diagonals in a Quadrilateral
  147. Bimedians in a Regular Tetrahedron [Java]
  148. Bisectal Circle [Java]
  149. Bisecting arcs
  150. Bisecting a shape
  151. Bisector of an imaginary angle may be real
  152. Bottema Shatters Japan’s Seclusion [Java]
  153. Bottema’s theorem [Java]
  154. Bottles in a Wine Rack [Java]
  155. Bounded Distance
  156. Brahmagupta’s Formula and Theorem
  157. Brahmagupta’s Theorem [Java]
  158. Brahmagupta-Mahavira Identities
  159. Brianchon in Ellipse [Java]
  160. Brianchon’s theorem [Java]
  161. Brick Wall Illusion [Java]
  162. Bride’s Chair [Java]
  163. Brief introduction to Rational Trigonometry
  164. Broken Chord Theorem [Java]
  165. Building a Bridge [Java]
  166. Building Bridges [Java]
  167. Bulging lines illusion [Java]
  168. Butterfly Theorem
  169. Cabart’s Collinearity
  170. Cantor Set and Function
  171. Cantor’s Theorem [Java]
  172. Carnot’s Theorem
  173. Carnot’s Theorem
  174. Carnot’s Theorem (Generalization of Wallace’s theorem) [Java]
  175. Carpet With a Hole [Java]
  176. Carpets Theorem [Java]
  177. Casey’s Theorem [Java]
  178. centers of similitude
  179. Centroid, a Characteristic Property Of
  180. Centroids in a Polygon [Java]
  181. Chasing Angles in Pascal’s Hexagon [Java]
  182. Circle, Isosceles Triangle and Fixed Point [Java]
  183. Circle Centers on Radical Axes [Java]
  184. Circle Chains on Napoleon Triangles [Java]
  185. Circle in a Square Inscribed in a Circle [Java]
  186. Circles on the sides of a triangle from the 2005 Austrian MO
  187. Circles Tangent to Circumcircle [Java]
  188. Circles with Tangent Diameters [Java]
  189. Ceva and Menelaus Meet on the Roads
  190. Ceva in Circumscribed Quadrilateral
  191. Ceva’s Theorem
  192. Ceva’s Theorem and Fibonacci Bamboozlement [Java]
  193. Cevian Cradle [Java]
  194. Cevian Nest [Java]
  195. Cevian Triangle [Java]
  196. Chain of Inscribed Circles [Java]
  197. Chaos, Emergence of [Java]
  198. Chasles’ Theorem
  199. Cherchez le quadrilatère cyclique [Java]
  200. Cherchez le quadrilatère cyclique II [Java]
  201. Chords, Concurrency and Orthic Triangle [Java]
  202. Chvatal’s Art Gallery Theorem [Java]
  203. Circle and Hyperbola as Lighthouse Curves
  204. Circle and Hyperbola as Lighthouse Curves II
  205. Circle Crosses Triangle [Java]
  206. Circle Concurrency and Spiral Similarity [Java]
  207. Circle Division by Euclid and Leonardo [Java]
  208. Circle of Similitude [Java]
  209. Circle through the Circumcenter [Java]
  210. Circle through the Incenter [Java]
  211. Circle through the Incenter And Antiparallels [Java]
  212. Circle-Stacking Theorem [Java]
  213. Circles And Parallels [Java]
  214. Circles and Semicircles in Rectangle [Java]
  215. Circles Cover a Quadrilateral [Java]
  216. Circles in a Circular Segment
  217. Circles in a Regular Polygon [Java]
  218. Circles in Morley’s Triangles [Java]
  219. Circles Lined on the Legs of a Right Triangle [Java]
  220. Circles On Cevians [Java]
  221. Circles through the Orthocenter [Java]
  222. Circles with Equal Collinear Chords [Java]
  223. Circular Poggendorff Illusion [Java]
  224. Circle Inscribed in a Circular Segment [Java]
  225. Circumcenter on Angle Bisector [Java]
  226. Circumcevian Triangle [Java]
  227. Classification of Quadrilaterals [Java]
  228. Cleavance Center [Java]
  229. Cleaver [Java]
  230. Clifford Algebras
  231. Clifford’s Chain [Java]
  232. Coaxal Circles on Perpendicular Bisector [Java]
  233. Coaxal Circles Theorem [Java]
  234. Collage Theorem [Java]
  235. Collinearity in Bicentric Quadrilaterals [Java]
  236. Collinearity via Concyclicity [Java]
  237. Collinearity with the Orthocenter [Java]
  238. Color Cycling on the Mandelbrot Set [Java]
  239. Coloring Plane with Three Colors
  240. Coloring Points in the Plane
  241. Common Chord and a Tangent [Java]
  242. Common Tangents to Two Circles I [Java]
  243. Common Tangents to Two Circles II [Java]
  244. Complete Quadrilateral [Java]
  245. Complex Numbers and Geometry
  246. Concurrency of the Altitudes in a Triangle - Trigonometric Proof
  247. Concyclicity in Rectangle [Java]
  248. Concurrency in the Intouch Triangle [Java]
  249. Concyclic Circumcenters: A Dynamic View [Java]
  250. Concyclic Points in a Triangle
  251. Concyclic Points in Bride’s Chair [Java]
  252. Concyclic Points in Equilateral Bumps [Java]
  253. Conic Sections
  254. Conic Sections as Loci of Points [Java]
  255. Conjugate Diameters in Ellipse [Java]
  256. Consecutive Isosceles Decomposition [Java]
  257. Constant Chord
  258. Constructing a triangle from its angle bisectors is in general impossible
  259. Construction of a Cyclic Quadrilateral
  260. Construction of a Square from the 2008 Austrian MO
  261. Construction of a Triangle from the Circumcenter, Orthocenter and Incenter
  262. Construction of a Triangle from Two Vertices and the Centroid
  263. Construction of an Oval - the Roman Way [GeoGebra]
  264. Construction of Conics from Pascal’s Theorem [Java]
  265. Construction of Regular Pentagon by H. W. Richmond
  266. Construction of the Angle Bisector [Java]
  267. Construction of the Perpendicular Bisector [Java]
  268. Construction problem [Java]
  269. Constructions Related To An Inaccessible Point
  270. Construction with the Compass Only
  271. Continuous Line Illusion [Java]
  272. Convex Polygon Is the Intersection of Half Planes [Java]
  273. Copernicus’ Theorem [Java]
  274. Construction with Optimization in Angle - Problem the 1979 USA Mathematical Olympiad
  275. cos(36°) = (1 + 5)/4
  276. Cosine Law
  277. Counting Triangles and Segments in a Polygon [Java]
  278. Cross Points in a Polygon [Java]
  279. Cross-Ratio
  280. Crossed-Lines Construction of Shapes of Constant Width [Java]
  281. Crossing Number of a Graph
  282. Cube Counting Ambiguity [Java]
  283. Curry’s Paradox [Java]
  284. Cut the Cone [Java]
  285. Cut the Cube [Java]
  286. Cut the Cylinder [Java]
  287. Cutting Squares
  288. Cutting Triangles off Regular Hexagon
  289. Cyclic Hexagon [Java]
  290. Cyclic Quadrilateral, Concurrent Circles and Collinear Points [Java]
  291. Cycloids [Java]
  292. Dancing Rectangles Model Auxetic Behavior [Java]
  293. Dancing Squares or a Hinged Plane Tessellation [Java]
  294. David Richeson’s Extention of an Old Japanses Theorem [Java]
  295. Day’s Sine Illusion [Java]
  296. A Decomposition of Isosceles Triangles into 4 Isosecles Triangles
  297. Delboeuf Illusion [Java]
  298. Delian Problem Solved [Java]
  299. Desargues’ Hexagon [GeoGebra]
  300. Desargues in the Bride’s Chair (with Pythagoras)
  301. Desargues’ Theorem [Java]
  302. Determination of π
  303. Diagonal Count [Java]
  304. Diagonals in a Cyclic Quadrilateral
  305. Diameters and Chords [Java]
  306. Directly Similar Figures [Java]
  307. Disappearing Lines puzzle [Java]
  308. Discovery of Duane DeTemple [Java]
  309. Dissection of a 10×13 Rectangle into Two Chessboards
  310. Dissection of a Vase [Java]
  311. Dissection of Cyclic Quadrilateral [Java]
  312. Dissection of Triangle into Rhombus [Java]
  313. Dissection of Triangle into Rhombus by Hubert Shutrick [Java]
  314. Distance
  315. Distance Between Projections
  316. Distance between the Orthocenter and Circumcenter
  317. Distance Formula
  318. Distorted Lines Illusion [Java]
  319. Divergent Spirals Illusion [Java]
  320. Divide a Circle into N Parts of Equal Area
  321. Divide a segment into n equal parts
  322. Dividing Evenly a Quadrilateral [Java]
  323. Dividing Evenly a Quadrilateral II [Java]
  324. Double Generation Theorem [Java]
  325. Drawing circles [Java]
  326. Droz-Farny Circles [Java]
  327. Droz-Farny Line Theorem [Java]
  328. Duplication of the Cube
  329. Dynamic Construction of Ellipse and Other Curves [Java]
  330. E. W. Dijkstra’s Proof of the Pythagorean Theorem [Java]
  331. Easy Construction of Bicentric Quadrilateral [Java]
  332. Easy Construction of Bicentric Quadrilateral II [Java]
  333. Egyptian Triangle By Paper Folding
  334. Eight Equal Tangents [Java]
  335. Eight Point Circle [Java]
  336. Ellipse in Arbelos [Java]
  337. Ellipse Between Two Circles [Java]
  338. Equal Angles in Two Circles [Java]
  339. Equal Angles in Two Circles II [Java]
  340. Equal Areas in Equilateral Triangle [Java]
  341. Equal Areas In Parallelogram à la Pythagoras [Java]
  342. Equal Chords in Crossing Circles [Java]
  343. Equal Circles, Medial Triangle and Orthocenter [Java]
  344. Equal Incircles Theorem [Java]
  345. Equichordal points: Just Do It [Java]
  346. Equidecomposition of a Rectangle and a Square [Java]
  347. Equidecomposition of Two Rectangles [Java]
  348. Equidecomposition of a Triangle and a Rectangle [Java]
  349. Equidecomposition of a Triangle and a Rectangle II [Java]
  350. Equidecomposition of Two Parallelograms [Java]
  351. Equilateral and 3-4-5 Triangles [Java]
  352. Equilateral Triangle on Parallel Lines [Java]
  353. Equilateral Triangle, Straight Line and Tangent Circles [Java]
  354. Equilateral Triangles and Incircles in a Square
  355. Equilateral Triangles on Diagonals of Antiequilic Quadrilateral [Java]
  356. Equilateral Triangles On Sides of a Parallelogram [Java]
  357. Equilateral Triangles On Sides of a Parallelogram II [Java]
  358. Equilateral Triangles on Sides of a Quadrilateral [Java]
  359. Equilic Quadrilateral I, a Flawed Proof [Java]
  360. Equilic Quadrilateral I, A Variation [Java]
  361. Equilic Quadrilateral II [Java]
  362. Erdös-Mordell Inequality
  363. Escher’s Theorem [Java]
  364. Euclid’s Fifth Postulate
  365. Euclid’s Elements Reference Page
  366. Euclid I.43 [Java]
  367. Euclid I.43 Extended [Java]
  368. Euclidean Construction of Center of Ellipse [Java]
  369. Euclidean Construction of Tangent to Ellipse
  370. Euler Line Cuts Off Equilateral Triangle [Java]
  371. Euler Line and 9-Point Circle [Java]
  372. Euler’s Formula and Poncelet Porism
  373. Every Parallelogram Is a Rectangle
  374. Existence of the Circumcenter [Java]
  375. Existence of the Euler Line: An Elementary Proof [Java]
  376. Existence of the Incenter: a Second Look
  377. Experimentation with Dynamic Geometry Software: An Example [Java]
  378. Exterior Angle Theorem - an appreciation
  379. External Angle Bisector [Java]
  380. Extouch Triangle in Poncelet Porism [Java]
  381. Extremal Problem in a Circular Segment
  382. The Extreme Principle
  383. Eyeball Theorem [Java]
  384. Fagnano’s problem [Java]
  385. Faulty Symmetry
  386. Fermat point and 9-point Centers [Java]
  387. Feuerbach’s Theorem: A Proof [Java]
  388. Find a Common Chord of Given Length [Java]
  389. Find the Angle! [Java]
  390. Finding a Parallelogram in 3D
  391. Finding Circle Center By Ruler Alone [Java]
  392. Five Concyclic Points [Java]
  393. Five Incircles in a Square [Java]
  394. Five Incircles Theorem [Java]
  395. Fixed Point in a Kite - Problem from the 1993 Asian Pacific MO
  396. Fixed Point in Isosceles and Equilateral Triangles
  397. Fixed Point of Circles Orthogonal to the Given One [Java]
  398. Fixed points from 2007 Irish Mathematical Olympiad
  399. Fleury’s Algorithm and Euler’s Paths and Cycles [Java]
  400. Folding Square in a Line through the Center [GeoGebra]
  401. Following the Hilbert Curve [Java]
  402. Foot of Altitude and Minimum Distance [Java]
  403. Focal Definition of Ellipse [Java]
  404. Focus and Directrix of Ellipse [Java]
  405. For every object there is a distance at which it looks its best.
  406. Ford’s touching circles
  407. Four Centroids and Parallels [Java]
  408. Four Circles In a Triangle [Java]
  409. Four Concurrent Lines in a Cyclic Quadrilateral [Java]
  410. Four Construction Problems [Java]
  411. Four Crossing Circles [Java]
  412. Four Geometric Problems of Antiquity
  413. Four Hinged Squares [Java]
  414. Four Incident Circles [Java]
  415. Four Incircles in Equilateral Triangle [Java]
  416. Four 9-Point Circles in a Quadrilateral [Java]
  417. Four Pedal Circles [Java]
  418. Four Touching Circles [Java]
  419. Four Touching Circles II [Java]
  420. Four Touching Circles III [Java]
  421. Four Triangles, One Circle [Java]
  422. Four Turtles (or Bugs, or Dogs, etc.) [Java]
  423. Fractal curves and dimension [Java]
  424. Friendly Kiepert’s Perspectors [Java]
  425. From Equilateral Triangle and Square to Golden Ratio
  426. From Foci to a Tangent in Ellipse [Java]
  427. Fundamental Theorem of 3-Bar Motion [Java]
  428. Fuss’ Theorem [Java]
  429. Generalizing a Romanian Olympiad Problem [Java]
  430. Geometric Illustration of a Convergent Series
  431. Geometric Mean Sangaku
  432. Geometric Meaning of the Geometric Mean
  433. Geometric Optimization from the Asian Pacific Mathematical Olympiad [Java]
  434. Geometric Proofs Of the Irrationality of Square Roots
  435. Geometry, Algebra, and Illustrations
  436. Gergonne and Medial Triangles Are Orthologic [Java]
  437. Gergonne and Soddy Lines Are Perpendicular [Java]
  438. Gergonne in Ellipse [Java]
  439. Gergonne and Van Obel Theorems
  440. Gion Shrine Problem
  441. Given Parabola, Find Axis
  442. Glide Reflection [Java]
  443. Golden Ratio in Geometry
  444. Golomb’s inductive proof of a tromino theorem [Java]
  445. Gothic Arc [Java]
  446. Graph Theory [Java]
  447. Grebe: from Ladies’ Diary to Carroll’s Pillow [Java]
  448. Grebe’s Theorem
  449. Greg Markowsky’s Problem for Parabola [Java]
  450. Griffiths Points in Cyclic Quadrilateral [Java]
  451. Griffiths’ Theorem
  452. Half Turn, Reflection in Point [Java]
  453. Halving a square
  454. Harmonic Mean in Geometry
  455. Harmonic Mean Sangaku [Java]
  456. Harmonic Ratio
  457. Harmonic Ratio in Complex Domain
  458. Hart Circle [Java]
  459. Hart’s Inversor [Java]
  460. Haruki’s Lemma [Java]
  461. Haruki’s Theorem
  462. Hausdorff Distance
  463. Hearty Munching on Cardioids [Java]
  464. Height/Width Interplay Illusion [Java]
  465. Helly’s Theorem
  466. Hering Illusion [Java]
  467. Hermann Grid Illusion [Java]
  468. Heron’s formula
  469. Heron’s Formula: a Proof [Java]
  470. Heron’s Problem [Java]
  471. Hexagon Parallel to Medial Triangle [Java]
  472. Hexagon Parallel to Orthic Triangle [Java]
  473. Hexagons in a Square
  474. Hinged Greek Cross Tessellation [Java]
  475. Hinged Squares [Java]
  476. Hippocrates’ Squaring of a Lune [Java]
  477. Hippocrates’ Squaring of Lunes
  478. Hjelmslev Theorem [Java]
  479. Homologous Lines under Three Spiral Similarities [Java]
  480. Homothety [Java]
  481. Homothety between In- and Excircles [Java]
  482. Hooper’s Paradox [Java]
  483. How Fast Does One’s Shadow Grow?
  484. Hunting Right Angles [Java]
  485. Huygens’ Problem
  486. Ian McGee’s Observation [Java]
  487. IMO 1998 Problem 1
  488. IMO 2004, Problem 5
  489. IMO 2006, Problem 1
  490. Impossible Fork [Java]
  491. Impossible Frame [Java]
  492. Importance of Occluders: Motion Binding [Java]
  493. In the Wasan Spirit
  494. Incenters in Cyclic Quadrilateral [Java]
  495. Incidence in Feuerbach’s Theorem [Java]
  496. Infinitude of Primes - A Topological Proof
  497. Inscribed and Central Angles in a Circle [Java]
  498. Inscribed Angles [Java]
  499. Inscriptible and Exscriptible Quadrilaterals [Java]
  500. Inscriptible Quadrilateral: An Illustration [Java]
  501. Internal Tangents to Three Circles [Java]
  502. Intersecting Chords Theorem
  503. Intersections of a Circle with the Four Quadrants [Java]
  504. Intouch Triangle in Poncelet Porism [Java]
  505. Invariance in Orthodiagonal Quadrilaterals [Java]
  506. Invariant Area Sweep Implies Pythagorean Theorem [Java]
  507. Inversion: Reflection in a Circle [Java]
  508. Involution on the Projective Line
  509. Is Every Trapezoid Parallelogram? [Java]
  510. Isosceles Tetrahedron
  511. Isogonal and Isotomic Conjugalities [Java]
  512. Isogonal Concurrency [Java]
  513. Isogonal image of the circumcircle [Java]
  514. Isoperimetric Theorem and Inequality
  515. Isosceles on the Sides of a Triangle [Java]
  516. Isotomy and Isogonality Hand-in-Hand [Java]
  517. Iterations and the Mandelbrot Set [Java]
  518. Iterations on a Circle Through Three Points [Java]
  519. Iterations on Euler Lines [Java]
  520. J. Casey’s Additions to Euclid I.47
  521. Jan Stevens’ proof of E. W. Dijkstra’s Generalization of the Pythagorean theorem
  522. Joachimsthal’s Notations
  523. John Sharp’s Paradox [Java]
  524. Johnson Circles [Java]
  525. Join Circles by Given Segment [Java]
  526. Joseph Keech in Bride’s Chair [Java]
  527. Judd Illusion [Java]
  528. Julia sets indexing [Java]
  529. Kanizsa Triangle [Java]
  530. Kiepert’s Theorem [Java]
  531. Knots [avi movie]
  532. Knots on a Torus [Java]
  533. Kürschak’s Tile and Theorem [Java]
  534. La Hire’s Theorem [Java]
  535. Langman’s Paradox [Java]
  536. Lean Napoleon’s Triangles [Java]
  537. Length of the diagonal of the unit square equals the square root of 2
  538. Lennes’ Polyhedron [Java]
  539. Leonardo’s Petals
  540. Lepidoptera of the Circles [Java]
  541. Line IO in Bicentric Quadrilaterals
  542. Line through a Point in an Angle
  543. Line, Circle, and Fixed Points [Java]
  544. Lipogrammatic Proof of the Pythagorean Theorem
  545. Loci of points in a cube
  546. Locus of Points in a Given Ratio to Two Points [Java]
  547. Longest segment
  548. Mach’s Conservation of Area and the Pythagorean Theorem
  549. MacLaurin’s Construction of Conics [Java]
  550. Maimed Cake [Java]
  551. Malfatti’s Problem [Java]
  552. Malfatti’s Problem, Hart’s Solution [Java]
  553. Mascheroni Construction of a Regular Pentagon
  554. Mathematical Droodles [Java]
  555. Mathematics in Pizzeria [Java]
  556. MathPro Logo [Java]
  557. Maximal Properties of the Pythagorean Relation
  558. Maximum Area Property of Equilateral Triangles
  559. Maximum Area Property of the Medial Triangle
  560. Maximum Perimeter Property of the Incircle {*}
  561. Maximum Ratio of Similarity - Belarus Olympiad Problem
  562. Maxwell’s Theorem [Java]
  563. May the Product of Planes Be a Sphere?
  564. Mechanical Proofs of the Pythagorean Theorem
  565. Median in Touching Circles [Java]
  566. Medians
  567. Medians in a Quadrilateral [Java]
  568. Medians in a Triangle Meet at the Center: a Second Look
  569. Meisters’ Two Ears Theorem
  570. Menelaus Theorem [Java]
  571. Metric Relations in a Triangle
  572. Midline in Similar Triangles [Java]
  573. Midpoints and Orthogonality in Isosceles Triangles [Java]
  574. Midpoints from Gergonne Triangle [Java]
  575. Midpoints of the Lines Joining In- and Excenters [Java]
  576. Mikhalev’s Octahedrons
  577. Minesweeper Theorem
  578. Miquel’s Point [Java]
  579. Miquel’s Point of a 4-line Via Spiral Similarity [Java]
  580. Miquel’s Theorem for Circles [Java]
  581. Mirror on the Wall [Java]
  582. Mirror property of the altitudes [Java]
  583. Mirror Property of Altitudes via Pascal’s Hexagram [Java]
  584. Mixtilinear Circles and Concurrence [Java]
  585. Moebius Strip [avi movie]
  586. Möbius’ Strip [Java]
  587. Models and Metamathematics
  588. Monge via Desargues [Java]
  589. Monge via Desargues II [Java]
  590. Morley Constellation [Java]
  591. Morley’s Miracle [Java]
  592. Morphing [Java]
  593. Müller-Lyer Illusion [Java]
  594. Multifaceted Cork [Java]
  595. Munching on Circles
  596. Munching on Inscribed Angles
  597. My logo
  598. Nagel Line [Java]
  599. Nagel point [Java]
  600. Nagel Point of the Medial Triangle [Java]
  601. Nagel’s Theorem [Java]
  602. Napoleon’s Theorem
  603. Necker Cube [Java]
  604. Neuberg Cubic [GeoGebra]
  605. Neuberg Sangaku [Java]
  606. Newly Born Pair of Siblings to Archimedes’ Twins [Java]
  607. Newton’s Construction of Conics [Java]
  608. Newton’s Theorem [Java]
  609. Nine Point Circle [Java]
  610. Nine Point Circle: an Elementary Proof [Java]
  611. No Equilateral Triangles, Please [Java]
  612. Nobbs’ Points, Gergonne Line [Java]
  613. Non-Eclidean Geometries
  614. No-Pedal Collinearity [Java]
  615. Occluded Wheel Illusion [Java]
  616. Olof Hanner’s Jigsaw Puzzle [Java]
  617. On the Difference of Areas
  618. One Sheet Hyperboloid [Java]
  619. Optimization in a Crooked Trapezoid
  620. Optimization Problem in Acute Angle [Java]
  621. Octagon in Parallelogram: Overlapping Regions [Java]
  622. Octagon in Parallelogram [Java]
  623. Optical Property of Ellipse [Java]
  624. Orbison illusion [Java]
  625. Orthocenter and Three Equal Circles [Java]
  626. Orthocenters
  627. Orthocenters of Two Triangles Sharing Circumcenter and Base [Java]
  628. Orthocentric System From Rectangle [Java]
  629. Orthodiagonal and Cyclic Quadrilaterals [Java]
  630. Orthogonality in Isosceles Triangles [Java]
  631. Orthologic Triangles in a Quadrilateral [Java]
  632. Orthopole [Java]
  633. Out of Pentagon Sangaku
  634. Packing shapes in a plane
  635. Pairs of Areas in Equilateral Triangle
  636. Pairs of Homologous Lines under Spiral Similarities [Java]
  637. Pairs of Incircles in a Quadrilateral [Java]
  638. Pantograph [Java]
  639. Paper Folding & Cutting Sangaku
  640. Paper Folding Geometry
  641. Paper strip activities [avi movies]
  642. Paper strip activities, two strips
  643. Pappus’ Generalization of Euclid I.47 [Java]
  644. Pappus’ Theorem [Java]
  645. Parabola [Java]
  646. Parabolic Mirror, Illustration [Java]
  647. Parabolic Mirror, Theory [Java]
  648. Parabolic Sieve of Prime Numbers
  649. Paragon [Java]
  650. Parahexagon [Java]
  651. Parallel Chords [Java]
  652. Parallel Chords II [Java]
  653. Parallel Chords in Ellipse [Java]
  654. Parallel Chords in Crossing Circles [Java]
  655. Parallel Lines in a Cyclic Quadrilateral [Java]
  656. Parallel Lines in a Quadrilateral [Java]
  657. Parallel Lines in a Triangle [GeoGebra]
  658. Parallelogram and Ellipses [GeoGebra]
  659. Parallelogram and Similar Triangles [Java]
  660. Parallelogram Illusion [Java]
  661. Parallelogram Iterations [Java]
  662. Parallelogram Law [Java]
  663. Parallelogram Law: A PWW [Java]
  664. Parallelogram Law: A Tessellation [Java]
  665. Partition of point sets in a plane
  666. Partitioning 3-Space with Circles
  667. Partitioning a Circle
  668. Pascal: Necessary and Sufficient [Java]
  669. Pascal in a Cyclic Quadrilateral [Java]
  670. Pascal Lines: Steiner and Kirkman Theorems [Java]
  671. Pascal Lines: Steiner and Kirkman Theorems II [Java]
  672. Pascal’s theorem [Java]
  673. Peacock’s Tail Sangaku
  674. Pedal Collinearities [Java]
  675. Pedal Polygons [Java]
  676. Pedal Triangle and Isogonal Conjugacy [Java]
  677. Pedoe’s Theorem [Java]
  678. Pencils of Cubics
  679. Pentagon, construction of
  680. Pentagon in a Semicircle [Java]
  681. Pentagon Proportions Sangaku
  682. Periodic patterns in sin(N) [Java]
  683. Perpendicular Bisectors in an Inscriptible Quadrilateral [Java]
  684. Perpendicular Bisectors in an Inscriptible Quadrilateral II [Java]
  685. Perpendicular lines in a cyclic quadrilateral from 1975 Canadian Mathematical Olympiad
  686. Phantom Circle and Recaptured Symmetry
  687. Pivot Theorem [Java]
  688. Plane Filling Curves [Java]
  689. Plane Isometries
  690. Plane Isometries As Complex Functions
  691. Plato’s Geometric Numbers: can you better Sam Loyd?
  692. Peaucellier Linkage [Java]
  693. Pick’s Theorem [Java]
  694. Poggendorff’s Illusion
  695. Point common to two similar rectangles [Java]
  696. Point in a Square
  697. Point on Bisector in Right Angle
  698. Points Generated by the Nine Points
  699. Polar Circle [Java]
  700. Poles and Polars [Java]
  701. Polygon in a Rectangle
  702. Polygons: formality and intuition [Java]
  703. Polygons Formed by Perpendicular Bisectors [Java]
  704. Pompeiu’s Theorem [Java]
  705. Poncelet’s Porism [Java]
  706. Poncelet’s Porism in Ellipses [Java]
  707. Poncelet’s Theorem [Java]
  708. Ponzo Illusion [Java]
  709. Ponzo Illusion II [Java]
  710. Possible or Impossible? [Java]
  711. Power of a point with respect to a circle
  712. Problem 4, 1975 USA Math Olympiad and the Radical Axis [Java]
  713. Problem 4, 1975 USA Math Olympiad and Isosceles Triangles [Java]
  714. Problem 4, 1975 USA Math Olympiad: Normals and Tangents [Java]
  715. Problem 4, 1975 USA Math Olympiad: Problem 4, 1975 USA Math Olympiad: The Final Touch[Java]
  716. Problem 4 from the IMO 2009
  717. Problem in an Equilateral Triangle [Java]
  718. Problem in an Equilateral Triangle II [Java]
  719. Problem in Three Squares [Java]
  720. Problem of Equal Steps [Java]
  721. Problem of Equal Steps II [Java]
  722. Problem on a Pentagonal Star
  723. Problem with angle bisector from 1975 Canadian Mathematical Olympiad
  724. Problem with equal angles from 1987 IberoAmerican Mathematical Olympiad
  725. Projections on Internal and External Angle Bisectors [Java]
  726. Projective Collinearity in a Quadrilateral [Java]
  727. Projective infinity
  728. Properties of the Circle of Similitude [Java]
  729. Properties of the Figures in Euclid I.47
  730. Property of 6-Parpolygon [Java]
  731. Property of Angle Bisectors [Java]
  732. Property of Angle Bisectors II [Java]
  733. Property of Circumscribed Quadrilaterals [Java]
  734. Property of Points Where In- and Excircles Touch a Triangle [Java]
  735. Property of Semicircles [Java]
  736. Proportions in Square
  737. Ptolemy’s Theorem
  738. Puzzles on graphs.
  739. Pythagoras’ Theorem (98 proofs)
  740. Pythagoras and Vecten Break Japan’s Isolation [Java]
  741. Pythagorean Triples [Java]
  742. Quadrature: A Child’s Play
  743. Quadrilateral from a Segment
  744. Quadrilateral With Equal Opposite Sides And Angles [Java]
  745. Quadrilateral with 3 Equal Sides [Java]
  746. Quadrilaterals Formed by Perpendicular Bisectors [Java]
  747. Quest for Paragon [Java]
  748. Radical Axis and Center, an Application [Java]
  749. Radical Axis of Circles Inscribed in a Circular Segment [Java]
  750. Radical Center [Java]
  751. Radius and Construction of a Mixtilinear Circle [Java]
  752. Radius of a Circle by Paper Folding [Java]
  753. Random Dot Stereograms [Java]
  754. Ratios and Orthogonality in a Triangle - Problem from the 2006 Romania MO
  755. Rational Points On a Circle
  756. Reflection In Line [Java]
  757. Reflections in Ellipse [Java]
  758. Reflections of a Line Through the Orthocenter [Java]
  759. Reflections of a Point on the Circumcircle [Java]
  760. Reflections of the Circumcenter
  761. Reflections of the Orthocenter [Java]
  762. Reflections of the Orthocenter II
  763. Regular Polygons in a Triangular Grid [Java]
  764. Regular Polyhedra - Platonic Solids
  765. Other Solids
  766. Regular Pentagon Inscribed in Circle by Paper Folding
  767. Regular polyhedra and Euler’s Theorem
  768. Relations in a Cyclic Polygon
  769. Remarkable Line in a Quadrilateral [Java]
  770. Remarkable Line in Cyclic Quadrilateral [Java]
  771. Rendezvous [Java]
  772. Reverse Spoke Illusion [Java]
  773. Revolving Circles Illusion [Java]
  774. Rhombus in a Cyclic Quadrilateral [Java]
  775. Riemann Sphere and Möbius Transformation
  776. Right Triangles on Sides of a Square [Java]
  777. Rotating Top Illusion [Java]
  778. Rotating Gears [Java]
  779. Rotation Transform [Java]
  780. Rouse Ball’s Fallacy [Java]
  781. Rusty Compass Construction of Equilateral Triangle [Java]
  782. Salinon: From Archimedes’ Book of Lemmas [Java]
  783. Sam Loyd’s Geometric Puzzle
  784. Sam Loyd’s Son’s Dissection [Java]
  785. Sangaku: Reflections and the Phenomenon
  786. Sangaku: Critique of My View and a Response
  787. Sangaku and The Egyptian Triangle
  788. Sangaku in a Square
  789. Sangaku Iterations, Is it Wasan? [Java]
  790. Sangaku with 8 Circles
  791. Sangaku with Angle between a Tangent and a Chord [Java]
  792. Sangaku with Quadratic Optimization
  793. Sangaku with Three Mixtilinear Circles [Java]
  794. Sangaku with Versines [Java]
  795. Sangakus with a Mixtilinear Circle [Java]
  796. Scintillating Grid Illusion [Java]
  797. Schwarz Lantern [Java]
  798. Secant Angles [Java]
  799. Secant Angles II [Java]
  800. Secant, Normal, Tangent [Java]
  801. Secant, Tangents and Orthogonality [Java]
  802. See-Saw Lemma [Java]
  803. Segment Trisection [Java]
  804. Segment Trisection Induced by Parallels to Medians [Java]
  805. Separating point sets with a circle
  806. Sequences of Touching Circles
  807. Serendipitous Beauty [Java]
  808. Serendipitous Proofs Of The Pythagorean Theorem
  809. Seven Circles Theorem [Java]
  810. Shaggy Dog Theorem
  811. Shapes in a lattice
  812. Shapes of constant width [Java]
  813. Shearing and Translation in Pythagorean Pants [Java]
  814. Shearing a Polygon into a Triangle of Equal Area [Java]
  815. Shearing Butterflies in Quadrilaterals [Java]
  816. Shepard’s Parallelogram Illusion [Java]
  817. Shifting Cube Illusion [Java]
  818. Shimmer Illusion [Java]
  819. Short Construction of the Geometric Mean
  820. Shortest Fence in a Quarter-Circle Pasture
  821. Shredding the torus
  822. Sierpinski Gasket By Common Trema Removal [Java]
  823. Sierpinski Gasket Via Chaos Game [Java]
  824. Sierpinski Gasket by Trema Removal [Java]
  825. Similar Triangles And More
  826. Similar Triangles on Sides of a Quadrilateral [Java]
  827. Similar Triangles on Sides and Diagonals of a Quadrilateral [Java]
  828. Simple dissection of an acute triangle
  829. Simple Quadrilaterals Tessellate the Plane [Java]
  830. Simson Line [Java]
  831. Simsons and 9-Point Circles in Cyclic Quadrilateral [Java]
  832. Simultaneous Contrast Illusion [Java]
  833. Simultaneous Contrast Illusion II [Java]
  834. Simultaneous Contrast: Koffka’s Illusion [Java]
  835. Simultaneous Contrast: White’s Illusion [Java]
  836. Simultaneous Contrast: White’s Illusion, V-Variant [Java]
  837. Simultaneous Diameters in Concurrent Circles [Java]
  838. Simultaneous Generalization of the Theorems of Ceva and Menelaus
  839. Sine and Cosine of 15 Degrees Angle
  840. Sine, Cosine, and Ptolemy’s Theorem
  841. Six Circles Theorem (Bui Quang Tuan) [Java]
  842. Six Circles Theorem (Elkies) [Java]
  843. Six Circles Theorem (Evelyn) [Java]
  844. Six Concyclic Points [Java]
  845. Six Concyclic Points II [Java]
  846. Six Incircles in an Equilateral Triangle [Java]
  847. Six Point Circle [Java]
  848. SixPoints, ThreeLines [Java]
  849. Soddy Circles and David Eppstein’s Centers [Java]
  850. Some Properties of Napoleon’s Configuration [Java]
  851. Sperner’s Lemma [Java]
  852. Spiral Similarity [Java]
  853. Square and Circle in a Gothic Cupola
  854. Square From Nowhere [Java]
  855. Square in a Circle Inscribed in a Square [Java]
  856. Square in a Right Triangle [Java]
  857. Square in Chair [Java]
  858. Square Inscribed in a Triangle [Java]
  859. Square Inscribed in a Triangle II [Java]
  860. Square, Similarity and Slopes [Java]
  861. Squares and Straight Tetrominoes
  862. Squares in Circles [Java]
  863. Squares in Semicircle and Circle
  864. Squares in Triangles
  865. Squares Inscribed In a Triangle III [Java]
  866. Squares on Sides of a Quadrilateral [Java]
  867. Square out of a Quadrilateral
  868. Squaring a circle
  869. Squaring a Rectangle [Java]
  870. Squinting Eyes Theorem [Java]
  871. SSA
  872. SSS
  873. Staircase Illusion [Java]
  874. Star of David [Java]
  875. Steiner’s Sangaku
  876. Steiner’s Theorem
  877. Stereographic Projection [Java]
  878. Stereographic Projection and Inversion
  879. Stereographic Projection and Radical Axes
  880. Stewart’s theorem
  881. Straight Edge Only Construction of Polar
  882. Structural Constellation [Java]
  883. Sum of Squares in Equilateral Triangle
  884. Sum of Squares of Distances to Vertices
  885. Symmedians
  886. Symmetries in a Triangle
  887. Tangency Point of Two Circles [Java]
  888. Tangent and Secant [Java]
  889. Tangent as a Radical Axis [Java]
  890. Tangent Circles and an Isosceles Triangle [Java]
  891. Tangent Circles in a Parallelogram [Java]
  892. Tangent in Concurrency [Java]
  893. Tangent Lines and Circles in Convex Quadrilateral [Java]
  894. Tangent to two circles
  895. Tangents and Diagonals in Cyclic Quadrilateral [Java]
  896. Tangents, Perpendiculars and Geometric Mean [Java]
  897. Tarski-Banach Decompositions
  898. Taylor circle [Java]
  899. Terquem’s Theorem [Java]
  900. Tesseract [Java]
  901. The 80-80-20 Triangle
  902. The Chaos Game: Address Space vs IFS [Java]
  903. The only rational triangle
  904. The Shape of the Colosseum [GeoGebra]
  905. Thébault’s Problem I [Java]
  906. Thébault’s Problem II [Java]
  907. Thébault’s Problem III [Java]
  908. Thébault’s Problem III, Proof [Java]
  909. Thébault’s Problem IV [Java]
  910. Theorem of Complete Quadrilateral [Java]
  911. Theorems of Ceva and Menelaus, an Illustrated Generalization [Java]
  912. Three Circles and Common Tangents
  913. Three Circles through the Incenter
  914. Three Concurrent Circles [Java]
  915. Three Congruent Circles by Reflection [Java]
  916. Three Congruent Circles by Reflection II [Java]
  917. Three Congruent Circles by Reflection III [Java]
  918. Three Equal Circles in a Semicircle
  919. Three Incircles In a Right Triangle [Java]
  920. Three Incircles in a Triangle [Java]
  921. Three Isosceles Triangles [Java]
  922. Three Orthogonal Circles Through Three Given Points [Java]
  923. Three Parallels in a Triangle [Java]
  924. Three Points on a Parabola [Java]
  925. Three Pyramids are Better Than Two [Java]
  926. Three regions sharing a common boundary
  927. Three Similar Polygons [Java]
  928. Three Similar Triangles [Java]
  929. Three Similar Triangles II [Java]
  930. Three Squares and Two Ellipses [Java]
  931. Three Tangent Circles [Java]
  932. Three Tangents, Three Chords in Ellipse [Java]
  933. Three Tangent Circles Sangaku
  934. Three Tangents Theorem [Java]
  935. Three Tangents, Three Secants [Java]
  936. Three Touching Circles [Java]
  937. Tiling a Chessboard with Dominoes
  938. Toothpick Construction of a Square [Java]
  939. Topological preliminaries
  940. Torque as a Mathematical Object
  941. Touching Circles [Java]
  942. Touching Circles and Concurrency [Java]
  943. Touching Circles with Given Centers II [Java]
  944. Train wheel quandary
  945. Transitivity in Action [Java]
  946. Translated Triangles [Java]
  947. Translation Transform [Java]
  948. Triangle ABC is right iff sin²A + sin²B + sin²C = 2
  949. Triangle Areas in a Parallelogram [Java]
  950. Triangle Areas in a Parallelogram II [Java]
  951. Triangle Classification [Java]
  952. Triangle constructions and facts [Java]
  953. Triangle Sliding on Two Straight Lines [Java]
  954. Triangles, Squares and Areas from Temple Geometry [Java]
  955. Triangles with Equal Area [Java]
  956. Triangular Billiard [Java]
  957. Triangulating Squares
  958. Trigonometric Form of Ceva’s Theorem [Java]
  959. Trigonometric Functions [Java]
  960. Trigonometric Identities with Arctangents
  961. Trigonometric Proof of the Pythagorean Theorem
  962. Trigonometry by Paper Folding
  963. Trisectable angle that is not constructible
  964. Tucker Circles [Java]
  965. Tucker Circles Through Homothety [Java]
  966. Twelve Matchsticks Area Puzzle
  967. Twisted Cord Illusion [Java]
  968. Two Altitudes, One Midpoint
  969. Two Arbelos, Two Chains
  970. Two Cevians and Proportions in a Triangle
  971. Two Circles and a Limit - Analytic Proof
  972. Two Circles and One More [Java]
  973. Two Circles and Two More [Java]
  974. Two Circles in a Parallelogram [Java]
  975. Two Circles Inscribed in a Parallelogram [Java]
  976. Two Circles in a Square [Java]
  977. Two Circles in a Square II [Java]
  978. Two Circles in an Angle [Java]
  979. Two Circles on Angle Bisector [Java]
  980. Two Circles on a Side of a Triangle [Java]
  981. Two Circumcircles And Two Pairs of Parallels [Java]
  982. Two Circumcircles in Triangle [Java]
  983. Two coins puzzle
  984. Two Congruent Circles by Reflection [Java]
  985. Two Color Coloring of the Plane [Java]
  986. Two Equilateral Triangles [Java]
  987. Two Homotheties in a Parallelogram [Java]
  988. Two Intersecting Circles [Java]
  989. Two Lines - Two Circles [Java]
  990. Two Pairs of Parallel Lines in a Triangle [GeoGebra]
  991. Two Parallels in a Triangle and One More [Java]
  992. Two Perpendiculars From a Point to a Line [Java]
  993. Two Planar Constructions Related to a 3D Problem
  994. Two Perspective Triangles
  995. Two Quadrilaterals [Java]
  996. Two Quadruplets of Tangent Circles [Java]
  997. Two Sangaku with Equal Incircles
  998. Two Simsons in a Triangle [Java]
  999. Two Squares and Another Square [Java]
  1000. Two Symmetric Triangles Are Directly Decomposable [Java]
  1001. Two Touching Circles [Java]
  1002. Two Triangles With Common Base and Altitude [Java]
  1003. Two Triples of Concurrent Circles [Java]
  1004. Two Triples of Similar Triangles [Java]
  1005. Unfolding Spirals Illusion [Java]
  1006. Urquhart’s Theorem
  1007. USA 1975 Problem 4
  1008. Use of Homogeneous Coordinates for Geometrical Calculations
  1009. Van Obel Theorem, Determinants, and Barycentric Coordinates
  1010. Van Schooten’s and Pompeiu’s Theorems [Java]
  1011. Van Schooten’s Locus Problem [Java]
  1012. Variations on the Theme of Tremas [Java]
  1013. Variations on the Theme of Tremas II [Java]
  1014. Varignon Parallelogram [Java]
  1015. Various Geometries
  1016. Vecten-Nikolin Hexagon [Java]
  1017. Vecten’s Collinearity [Java]
  1018. Vecten’s Mesh [Java]
  1019. Vector Addition and Subtraction [Flash]
  1020. Vector product
  1021. Versatile Theorem [Java]
  1022. Vertical and Horizontal Dominoes on a Chessboard
  1023. Viewing a Statue
  1024. Viviani’s Theorem [Java]
  1025. Viviani’s Theorem II [Java]
  1026. Viviani in Isosceles Triangle [Java]
  1027. Viviani by Inversion
  1028. Viviani by Vectors
  1029. Volume of Sphere and Volume of an Ellipsoid
  1030. Volumes of Two Pyramids
  1031. Wallace-Bolyai-Gerwien Theorem
  1032. Wallace’s Theorem [Java]
  1033. Watt’s and Chebyshev’s Linkage [Java]
  1034. Weird curves bound normal areas
  1035. Weitzenböck Inequality [Java]
  1036. What does one see inside a spherical mirror? [Java]
  1037. What Is Analytic Geometry?
  1038. What Is Angle?
  1039. What Is Area?
  1040. What Is Ellipse?
  1041. What Is Line?
  1042. What Is Point?
  1043. When a Triangle is Equilateral [Java]
  1044. When index equals content [Java]
  1045. Why a Diameter Is the Longest Chord?
  1046. William Wallace’s Proof of the Butterfly Theorem
  1047. Wittenbauer’s Parallelogram [Java]
  1048. Wundt Illusion [Java]
  1049. Wundt Block Illusion [Java]
  1050. Y. Sawayama’s Lemma [Java]
  1051. Y. Sawayama’s Theorem
  1052. Zaslavsky’s theorem [Java]
  1053. Zöllner’s Illusion [Java]

Crux Maths Problems

Sunday, October 7th, 2012

This journal (also called CRUX with MAYHEM) is an internationally respected source of unique and challenging mathematical problems published by the CMS. Designed primarily for the secondary and undergraduate levels, and also containing some pre-secondary material, it has been referred to as “the best problem solving journal in the world”. All the problems and solutions are fully peer-reviewed for clarity, completeness and rigour by academic and professional mathematicians. Crux includes an “Olympiad Corner” which is particularly helpful for students preparing for math competitions.

Địa chỉ chính thức: https://cms.math.ca/crux/

Link tải một số tài liệu: tại đây

  • Năm 1996
  • Năm 1997
  • Năm 1998
  • Năm 1999
  • Năm 2000
  • Năm 2001
  • Năm 2002
  • Năm 2003
  • Năm 2004
  • Năm 2005
  • Willian Putnam Problems

    Sunday, October 7th, 2012
    Year Problems Solutions Winners Scores
    2011 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    2010 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    2009 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    2008 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    2007 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    2006 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    2005 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    2004 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    2003 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    2002 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    2001 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    2000 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    1999 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    1998 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    1997 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    1996 TeX DVI PS PDF TeX DVI PS PDF HTML
    1995 TeX DVI PS PDF TeX DVI PS PDF HTML HTML
    1994 TeX DVI PS PDF HTML HTML
    1993 TeX DVI PS PDF
    1992 TeX DVI PS PDF
    1991 TeX DVI PS PDF
    1990 TeX DVI PS PDF
    1989 TeX DVI PS PDF
    1988 TeX DVI PS PDF
    1987 TeX DVI PS PDF
    1986 TeX DVI PS PDF
    1985 TeX DVI PS PDF

    Phương pháp đại số - Phương pháp gien

    Sunday, October 7th, 2012

    Phương pháp đại số (Phương pháp gien)

    Tham khảo bài viết tại đây

    Nếu từ một nghiệm của phương trình đã cho ta có quy tắc để xây dựng ra một nghiệm mới thì quy tắc đó chính là gien. Phương pháp gien là phương pháp dựa vào gien để tìm tất cả các nghiệm của phương trình đã cho từ các nghiệm cơ sở. Để tìm các nghiệm cơ sở, ta áp dụng bước lùi, tức là quy tắc ngược của quy tắc tiến nói trên. Minh họa tốt nhất cho ý tưởng này là phương trình Pell và phương trình Markow. Ta bắt đầu bằng phương trình Pell.

    Phương trình Pell cổ điển là phương trình dạng, Dvô tỷ nên nếu S0là tập hợp tất cả các nghiệm cơ sở. Ta có định lý quan trọng sau:

    Định lý 3. Với mọi D, k ta có  S0, bằng cách đi ngược xuống bằng công thức  x’ = ax ồ Dby, y’ = ay ồ bx  ta luôn có thể đi đến một nghiệm cơ sở của (4). Như vậy, với định lý trên, phương trình dạng Pell đã được giải quyết hòan tòan. Dưới đây chúng ta xem xét một ví dụ:

    Ví dụ 1. Tìm tất cả các nghiệm nguyên dương của phương trình x25y2=4(1).

    Lời giải: Bằng phép thử tuần tự, ta tìm được nghiệm cơ sở của phương trình x25y2=1là (9, 4). Theo phép chứng minh định lý 3, nghiệm cơ sở của (1) thỏa mãn
    x24.5.42,y2(4.5.42+4)/5  =>x<17,y<9
    Dùng phép thử tuần tự, ta tìm được hai nghiệm cơ sở là (1, 1) và (11, 5). Từ hai nghiệm này, bằng công thức
    x’ = 9x + 20y, y’ = 4x + 9y
    ta tìm được tất cả các nghiệm của (1).

    Với phép giải phương trình dạng Pell, trên thực tế ta đã có thể giải tất cả các phương trình Diophant bậc 2, tức là phương trình dạng

    ax2+bxy+cy2+dx+ey+f=0

    Dựa vào lý thuyết đường cong bậc 2, ta có thể đưa phương trình trên về một trong các dạng chính sau
    + Dạng ellip:  ax2+by2=c (a, b, c > 0) ồ có hữu hạn nghiệm, giải bằng phương pháp thử và sai
    + Dạng parabol: ax2+by+c=0- giải bằng đồng dư bậc 2
    + Dạng hypebol: ax2by2=c- phương trình dạng Pell
    Ngòai ra còn có các dạng suy biến như hai đường thẳng cắt nhau, hai đường thẳng song song, ellip ảo … Dưới đây, ta xét một ví dụ áp dụng:

    Ví dụ 2: Tìm tất cả các cặp số nguyên dương (m, n) thỏa mãn phương trình
    \displaystyle{m(m+1) + n(n+1) = 3mn#}
    Lời giải: Xét phương trình đã cho như phương trình bậc 2 theo m
    m2(3n1)m+n(n+1)=0
    Phương trình này có nghiệm nguyên dương khi và chỉ khi   :Rightarrow là số chính phương, tức là
    (3n1)24n(n+1)=y2<=>y25(n1)2=4ta thu được phương trình dạng Pell mà ta đã biết cách giải.

    [i](Trích bài giảng về phương trình Diophant) 

    Maths problems

    Saturday, October 6th, 2012

    Article printed from Dong Thap in South Vietnam: http://dongthapvietnam.informe.com/blog

    URL to article: http://dongthapvietnam.informe.com/blog/trang-luyện-tập-mon-toan/

    URLs in this post:
    [1] Inequalities Theorems and Formulas forum: http://www.artofproblemsolving.com/Forum/viewforum.php?f=55
    [2] Useful Inequalitieshttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=7942
    [3] Polynomials and Polynomial Inequalities (Graduate Texts in Mathematics)http://www.amazon.com/Polynomials-Polynomial-Inequalities-Graduate-Mathematics/dp/0387945091/ref=sr_
    1_3?s=books&ie=UTF8&qid=1304702384&sr=1-3

    [4] Geometry of Polynomialshttp://www.amazon.com/Geometry-Polynomials-Mathematical-Surveys-Monographs/dp/0821815032/ref=cm_cr_p
    r_product_top

    [5] Polynomialshttp://www.amazon.com/Polynomials-Problem-Books-Mathematics-Barbeau/dp/0387406271/ref=cm_cr_pr_produ
    ct_top

    [6] Solving Polynomial Equations: Foundations, Algorithms, and Applications: http://www.amazon.com/Solving-Polynomial-Equations-Foundations-Applications/dp/3642063616/ref=sr_1_5
    ?s=books&ie=UTF8&qid=1304702384&sr=1-5

    [7] Number Theory: http://www.artofproblemsolving.com/Resources/Papers/SatoNT.pdf
    [8] download herehttp://www.artofproblemsolving.com/Forum/download/file.php?id=32932
    [9] download herehttp://www.artofproblemsolving.com/Forum/download/file.php?id=32931
    [10] download herehttp://www.artofproblemsolving.com/Forum/download/file.php?id=32930
    [11] download herehttp://www.artofproblemsolving.com/Forum/download/file.php?id=32929
    [12] download herehttp://www.artofproblemsolving.com/Forum/download/file.php?id=33191
    [13] herel: http://www-math.mit.edu/~kedlaya/geometryunbound/
    [14] All Volumeshttp://www.4shared.com/file/9Ay946kL/Forum_Geometricorum_-_All_volu.html
    [15] First Volumehttp://www.4shared.com/document/Nm86Tfiv/Volume_1_-_FORUM_GEOMETRICORUM.html
    [16] Second Volumehttp://www.4shared.com/document/-Kg5kOx8/Volume_2_-_FORUM_GEOMETRICORUM.html
    [17] Third Volumehttp://www.4shared.com/document/dcCAx9Vp/Volume_3_-_FORUM_GEOMETRICORUM.html
    [18] Fourth Volumehttp://www.4shared.com/document/tNqXyU1F/Volume_4_-_FORUM_GEOMETRICORUM.html
    [19] Fifth Volumehttp://www.4shared.com/document/NFT7Irys/Volume_5_-_FORUM_GEOMETRICORUM.html
    [20] Sixth Volumehttp://www.4shared.com/document/NqofMPAt/Volume_6_-_FORUM_GEOMETRICORUM.html
    [21] Seventh Volumehttp://www.4shared.com/document/LllTHOec/Volume_7_-_FORUM_GEOMETRICORUM.html
    [22] Eighth Volumehttp://www.4shared.com/document/yIx2G_2X/Volume_8_-_FORUM_GEOMETRICORUM.html
    [23] Ninth Volumehttp://www.4shared.com/document/3u061hi3/Volume_9_-_FORUM_GEOMETRICORUM.html
    [24] Thishttp://www.cip.ifi.lmu.de/~grinberg/faq.html#books
    [25] Darij Grinberghttp://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=432
    [26] Darij Grinberg’s whole site downloadhttp://sites.google.com/site/darijgrinberg/website
    [27] websitehttp://www.cip.ifi.lmu.de/~grinberg/
    [28] 100 Inequality Problems Proposed by Vasc and arqadyhttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=385236
    [29] 115 Algebra Problemshttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=406530
    [30] 100 Nice Polynomial Problems (With Solutions)http://www.artofproblemsolving.com/Forum/viewtopic.php?t=397768
    [31] 100 Functional Equations Problems (With Solutions)http://www.artofproblemsolving.com/Forum/viewtopic.php?t=411461
    [32] Trigonometry Problems Collectionhttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=391940
    [33] 567 Nice And Hard Inequalitieshttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=386343
    [34] Inequalities From 2007 and 2008 Competitions Around The Worldhttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=398739
    [35] A Collection of Limitshttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=398915
    [36] Pain rinneganhttp://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=61082
    [37] Technical Analysis of Three Variable Inequalitieshttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=407914
    [38] 1220 Number Theory Problems (With Sources)http://www.artofproblemsolving.com/Forum/viewtopic.php?t=492767
    [39] Lifting the Exponent Lemma (LTE)http://www.artofproblemsolving.com/Forum/viewtopic.php?t=401494
    [40] Solving Diophantine Equationshttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=410293
    [41] lamphonghttp://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=94338
    [42] Several Things About Sum of Squareshttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=408084
    [43] lamphonghttp://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=94338
    [44] Some Own Problems In Number Theoryhttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=399644
    [45] mathmdmbhttp://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=85314
    [46] Primitive Roots, Order, and Quadratic Residueshttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=399036
    [47] mathmdmbhttp://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=85314
    [48] Number Theory Marathon Problemshttp://www.artofproblemsolving.com/Forum/viewtopic.php?t=396930
    [49] M4RI0http://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=16383
    [50] 150 Nice Geometry Problems (With Solutions)http://www.artofproblemsolving.com/Forum/viewtopic.php?t=385682
    [51] Darij Grinberg’s whole site downloadhttp://sites.google.com/site/darijgrinberg/website
    [52] websitehttp://www.cip.ifi.lmu.de/~grinberg/
    [53] 100 Combinatorics Problems (With Sources)http://www.artofproblemsolving.com/Forum/viewtopic.php?t=412125
    [54] AoPS Resources Page Problems (IMO and ShortLists Added)http://www.artofproblemsolving.com/Forum/viewtopic.php?t=361972
    [55] download herehttp://www.artofproblemsolving.com/Forum/download/file.php?id=33011
    [56] 150 nice geometry problem: http://www.artofproblemsolving.com/Forum/download/file.php?id=30649
    [57] Bài 1: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=410614
    [58] Bài 2: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=320016
    [59] Bài 3: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=320299


    Bồi dưỡng hè - Chủ Nhật, 12/08/2012, Tuoitreonline

    Sunday, August 12th, 2012

    TT - Mỗi năm đến hè giáo viên chúng tôi được bồi dưỡng. Bồi dưỡng chuyên môn, bồi dưỡng chính trị, bồi dưỡng đổi mới phương pháp giảng dạy…

     

    Minh họa: NGUYỄN NGỌC THUẦN

    Sau đây là một buổi bồi dưỡng điển hình.

    Tám giờ báo cáo viên đến lớp. Tóc hoa râm, kính trắng, răng cửa rụng hai chiếc, trịnh trọng mở laptop. Trong khi kỹ thuật viên chỉnh mic, đèn chiếu, màn ảnh… báo cáo viên vô PowerPoint, màn ảnh hiện lên tấm hình ông trong hội thảo có cái tên dài lum bum: “Nghiên cứu khoa học sư phạm thực nghiệm bậc trung học phổ thông. Dự án hợp tác Việt - Áo”. Báo cáo viên thổi phù phù vô mic thử âm thanh và bắt đầu:

    - Chào các đồng chí, à chào quý thầy cô thì đúng hơn, mình dân sư phạm mà. Chúng ta lại gặp nhau vào dịp cuối hè, chúng ta hết nghỉ hè rồi phải không, sắp vô năm học mới rồi. Hôm nay tôi sẽ bồi dưỡng quý thầy cô chuyên đề nghiên cứu khoa học sư phạm thực nghiệm, hiệu quả lắm các đồng chí. Thiệt ra tôi cũng như quý thầy cô thôi, vấn đề ở chỗ tôi được phân công đi tập huấn về báo cáo lại, nếu có gì sơ suất mong quý thầy cô góp ý hoặc thông cảm, hoặc bỏ qua.

    Báo cáo viên ngó lên màn hình, tiếp:

    - Đây là tôi trong buổi thảo luận với mấy anh em khắp các tỉnh thành. Vừa rồi tôi học một tuần ở Hà Nội về cái chuyên đề này, trời nóng khiếp, nhưng tôi sẽ tóm gọn báo cáo lại cho các đồng chí, à cho quý thầy cô, hai ngày thôi.

    Một nam giáo viên la lên: “Một ngày thầy ơi”.

    Báo cáo viên nheo mắt:

    - Nói hai ngày nhưng mai trúng thứ bảy rồi, chiều ta nghỉ sớm coi như chỉ ngày rưỡi thôi, đúng không. Trước hết tôi nói về giờ giấc với quý thầy cô. Mình phải nghiêm túc, kiên quyết đúng giờ, dứt khoát không dây thun các đồng chí. Sáng, ta sẽ bắt đầu từ bảy giờ rưỡi.

    Một nữ giáo viên la lên: “Tám giờ đi thầy”.

    - Thôi thôi, ta bắt đầu bảy rưỡi làm tới mười rưỡi là ba tiếng, nghỉ giải lao ba mươi phút, còn lại hai tiếng rưỡi chớ mấy. Ta nghỉ giải lao lúc chín giờ, chín rưỡi ta vào lớp ngay, không ai được dây dưa đứng ngoài hành lang. Buổi chiều ta bắt đầu lúc một giờ rưỡi.

    Lại một giáo viên la lên: “Hai giờ đi thầy”.

    - Được rồi, ta làm sớm nghỉ sớm, từ một rưỡi đến bốn rưỡi là ba tiếng, nghỉ giải lao ba mươi phút, còn hai tiếng rưỡi chớ mấy. Ta nghỉ giải lao lúc ba giờ, ba rưỡi ta vào lớp ngay quý thầy cô ạ. Vâng, chúng ta vừa thống nhất xong giờ giấc, quý thầy cô nhớ nhé. Sao không thấy ai ghi vô sổ tay để ghi nhớ nhè? (cười)

    Báo cáo viên rê chuột, hiện lên tấm hình giáo sư Đăng Trình nép bên cạnh là báo cáo viên:

    - Ông này giỏi lắm các đồng chí, à quý thầy cô. Phía Áo họ giao hết cho giáo sư dịch giáo trình này. Tụi tôi học căng lắm, sáng học chiều thảo luận, chiều học sáng thảo luận suốt một tuần không đi chơi đâu được. Mình đại diện cho tỉnh phải học tập nghiên cứu cho tốt, chớ đâu để anh em tỉnh khác nó khinh mình được. Cuối đợt làm bài kiểm tra, tôi chọn đề tài “giảm thiểu tối đa tỉ lệ học sinh bỏ học”, tỉnh mình nổi tiếng bỏ học các đồng chí à, đề tài được giáo sư Đăng Trình chấm điểm A.

    Báo cáo viên chạy đề tài vun vút trên màn ảnh, phủ dụ:

    - Ngày mai tôi sẽ photo nó, phát cho các đồng chí nghiên cứu.

    Một giáo viên la lên: “Nghỉ giải lao thầy ơi, chín giờ rồi”.

    Báo cáo viên xem đồng hồ, chín giờ kém mười:

    - À, nghỉ giải lao nhè.

    Dù báo cáo viên căn dặn chín rưỡi vô lại lớp nhưng đến mười giờ kém mới không còn giáo viên nào đứng ở hành lang.

    - Xin các đồng chí ổn định trật tự. Bây giờ tôi sẽ nói tổng quan về nghiên cứu khoa học sư phạm thực nghiệm. Quý thầy cô im lặng chút đi. Nghiên cứu khoa học sư phạm thực nghiệm tất nhiên là khác hẳn viết sáng kiến kinh nghiệm rồi. À, xin bật mí cho các đồng chí biết, năm tới giám đốc sở yêu cầu giáo viên viết đề tài về nghiên cứu khoa học sư phạm thực nghiệm thay cho viết sáng kiến kinh nghiệm đó. Sáng kiến kinh nghiệm luôn mang tính chủ quan và cá nhân các đồng chí à. Còn nghiên cứu khoa học sư phạm thực nghiệm mang tính khách quan, tương tác với các đối tượng.

    Lớp học là hội trường lớn chứa hơn hai trăm giáo viên, mạnh ai nấy nói chuyện. Họ nói về giá xăng, về thực phẩm độc hại, về Trung Quốc, về con cái làm mất điện thoại hoặc máy tính, về dạy thêm bị học sinh chạy làng… inh ỏi cả hội trường.

    Báo cáo viên hắng giọng vô mic:

    - Xin các thầy cô im lặng giùm.

    Báo cáo viên trễ kính xuống sống mũi, ngán ngẩm. Tiếng hát át tiếng bom thôi, báo cáo viên dùng phương pháp phát vấn “thế nào và ra sao” gào vô mic:

    - Cho tôi được tiếp tục quý thầy cô nhé. E hèm, thế nào là nghiên cứu khoa học sư phạm thực nghiệm, tại sao phải nghiên cứu khoa học sư phạm thực nghiệm? Nghiên cứu ra sao, kết quả thế nào hả các đồng chí? Tác dụng của nghiên cứu khoa học sư phạm thực nghiệm đến đâu, tại sao phải nghiên cứu nó? Dùng phương pháp nào, định tính hay định lượng? Chọn đề tài thế nào, khung nghiên cứu ra sao? Đối tượng nghiên cứu là cái gì, tại sao chọn đối tượng đó? Các đồng chí cần phải trả lời rốt ráo mấy câu hỏi vừa nêu. Mục đích đặt ra là gì, căn cứ vào đâu, quy trình thế nào, dữ liệu đầu vào đầu ra nói lên điều gì? Các đồng chí cần phải nắm bắt thật cụ thể, sát sao, quyết liệt.

    Chưa đến mười rưỡi nhưng một giáo viên cứng tuổi la lên: “Hết giờ thầy ơi”.

    - Hết giờ rồi à, nghỉ nhè. Chiều nay quý thầy cô có mặt đúng một rưỡi nhé.

    Sau mỗi đợt bồi dưỡng như vậy, tôi làm trắc nghiệm: “Bạn tiếp thu tốt chứ?” a) cười, b) lắc đầu, c) gật đầu, d) phương án khác. Kết quả thu được: 30% chọn phương án a, 70% chọn phương án b.

    Truyện 1.177 chữ của PHÙNG HI

    boiduonghe

    THPT Lý Tự Trọng - Nha Trang

    Thursday, August 2nd, 2012

    CIMG2687

    CIMG2648

    CIMG2649

    CIMG2650

    CIMG2651

    CIMG2652

    CIMG2653

    CIMG2654

    CIMG2655

    CIMG2656

    CIMG2657

    CIMG2658

    CIMG2659

    CIMG2660

    CIMG2661

    CIMG2662

    CIMG2663

    CIMG2664

    CIMG2665

    CIMG2666

    CIMG2667

    CIMG2668

    CIMG2669

    CIMG2670

    CIMG2671

    CIMG2672

    CIMG2673

    CIMG2674

    CIMG2675

    CIMG2676

    CIMG2677

    CIMG2678

    CIMG2679

    CIMG2680

    CIMG2681

    CIMG2682

    CIMG2683

    CIMG2684

    CIMG2685

    CIMG2686

    upload picture

    Dong Thap - Viet Nam

    Wednesday, September 10th, 2008

    Trang thông tin Đồng Tháp - Việt Nam

     


    Tạo ảnh động ghép hình

    Ảnh động hoa sen Tham khảo