مساله2

Prove the following: if, in a triangle ABC,  mÐ BAO = mÐ OAC = 20°, mÐ ACO = 10°,and mÐ OCB = 30°, then mÐ OBC = 80°.

مساله

Prove the following: if, in a triangle ABC, median BD is such that mÐ A = mÐ DBC, and mÐ ADB = 45°, then mÐ A = 30°.

مسائل وقضایای هندسه

Table of Content

1. Triangles. Theorems and problems about triangles.

2. Circles. Theorems and problems about circles.

3. Triangle Centers

4. Archimedes' Book of Lemmas

5. Quadrilaterals

6. Butterfly Theorem. See also: Butterfly Theorem Puzzle

7. Carnot's Theorem

8. Ceva's Theorem

9. Clifford's Circles Chain Theorems

10. Equal Incircles Theorem

11. Equilic Quadrilateral.

12. Eyeball Theorem. Illustration

13. Gergonne Point Theorem

14. Heron's Formula. Key facts and a purely geometric step-by-step proof.

15. Johnson's Theorem

16. Kurschak's Tile & Theorem

17. Langley Problem: 20° Isosceles Triangle Adventitious angles.

18. Menelaus' Theorem

19. Miquel's Pentagram. Proof

20. Miquel's Pentagram. Dynamic Geometry Cabri, GSP, Cinderella, C.a.R.

21. Monge & d'Alembert Three Circles Theorem I - Dynamic Geometry Requires Java applet 1.3 or higher.

22. Monge & d'Alembert Three Circles Theorem II - Dynamic Geometry Requires Java applet 1.3 or higher.

23. Morley's Theorem

24. Napoleon's Theorem

25. Nagel Point Theorem. See also: Nagel Point Flowchart Proof, Nagel Point Puzzle.

26. Newton's Theorem: Newton's Line

27. Parallelogram with Squares theorem.

28. Pentagons and Pentagrams. Menelaus and Collinearity

29. Platonic Solids. Interactive animation.

30. Poncelet's Theorem

31. Proposed Problems about congruence of line segments, angles, and triangles. Level: High School, SAT Prep, College geometry.

32. Sangaku Problem

33. Sangaku Problem 2

34. Sangaku Problem 3

35. Semiperimeter and excircles of a triangle

36. Semiperimeter and incircle of a triangle

37. Semiperimeter, incircle and excircles of a triangle

38. Seven Circles Theorem

39. Simson Line

40. Simson Line: Angle between two Simson lines

41. Steiner's Theorem

42. 100° Isosceles Triangle Problem

43. Triangles

44. Triangle with the bisectors of the exterior angles. Collinearity

45. Triangle & squares. Fifteen theorems, visual illustrations.

46. Triangle with Squares 0 Two squares.

47. Triangle with Squares 1 Two squares.

48. Triangle with Squares 2 Two squares.

49. Triangle with Squares 3 Three squares.

50. Triangle with Squares 4 Finsler-Hadwiger Theorem.

51. Triangle with Squares 5 Two squares, median and altitude.

52. Triangle with Squares 6 Four squares.

53. Triangle Centers

54. Van Aubel's Theorem Quadrilateral with Squares

55. Varignon and Wittenbauer parallelograms

56. Adjacent Angles and Five Bisectors