The triangle is an amazingly rich structure, with a maltitude of relations between its elements. Below I shall be using standard notations: a, b, c, for the side length, p for the semiperimeter, etc.

1. Ceva's Theorem
2. Law of Cosines
3. Law of Sines
4. Menelaus' Theorem
5. Stewart's Theorem
6. Van Obel Theorem
7. Carnot's Theorem
8. 2·S = a·h_a = b·h_b = c·h_c.
9. S² = p(p - a)(p - b)(p - c).
10. h_a² = 4S²/a² = 4p(p - a)(p - b)(p - c)/a², etc.
11. S = rp.
12. r² = p^-1(p - a)(p - b)(p - c).
13. 1/r = 1/h_a + 1/h_b + 1/h_c.
14. h_a = c·sin(B) = b·sin(C), etc.
15. 2·S = bc·sin(A) = ac·sin(B) = ab·sin(C).
16. tan(A/2) = r/(p - a), etc.
17. S = r_a(p - a), etc.
18. sin²(A/2) = (p - b)(p - c) / bc, etc.
19. cos²(A/2) = p(p - a) / bc, etc.
20. AI = r/sin(A/2), etc.
21. AI² = (p - a)bc/p, etc.
22. AI² = bc·tan(B/2)·tan(C/2), etc.
23. 1/r = 1/r_a + 1/r_b + 1/r_c.
24. r_a + r_b + r_c = r + 4R.
25. r_ar_br_c = pS.
26. r r_ar_br_c = S².
27. r_a² = p(p - a)^-1(p - b)(p - c), etc.
28. l_a² = 4p(p - a)bc(b + c)^-2, etc.
29. m_a² = (b² + c²)/2 - a²/4, etc.
30. abc = 4RS.
31. bc = 2Rh_a.
32. OI² = R² - 2Rr.
33. r + r_a + r_b - r_c = 4Rcos(C).
34. r_a + r_b = 4Rcos²(C/2).
35. p = 4Rcos(A/2)·cos(B/2)·cos(C/2).
36. p - a = 4Rcos(A/2)·sin(B/2)·sin(C/2).
37. S = 2R²sin(A)·sin(B)·sin(C).
38. r = 4Rsin(A/2)·sin(B/2)·sin(C/2).
39. 2vers(A)vers(B)vers(C) = r²/R².
40. cot(A/2) + cot(B/2) + cot(C/2) = cot(A/2)·cot(B/2)·cot(C/2).
41. vers(A)vers(B)vers(C) = 8·AI·BI·CI/R³
42. rR = abc / 4p
43. AH = 2R·|cos(A)|

Copyright © 1996-2009 Alexander Bogomolny