Metric Relations in a Triangle

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.

(Note that the are also very useful inequalities between the elements, and purely trigonometric identities between the angles of a triangle.)

  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·ha = b·hb = c·hc.
  9. 2·S = ab sin(C) = ac·sin(B) = bc·sin(A).
  10. S² = p(p - a)(p - b)(p - c).
  11. ha² = 4S²/a² = 4p(p - a)(p - b)(p - c)/a², etc.
  12. S = rp.
  13. r² = p-1(p - a)(p - b)(p - c).
  14. 1/r = 1/ha + 1/hb + 1/hc.
  15. ha = c·sin(B) = b·sin(C), etc.
  16. cot(A/2) = (p - a)/r, etc.
  17. S = ra(p - a), etc.
  18. sin²(A/2) = (p - b)(p - c) / bc, etc.
  19. cos²(A/2) = p(p - a) / bc, etc.
  20. tan²(A/2) = (p - b)(p - c) / p(p - a), etc.
  21. cos²[(C-B)/2] = [(b+c)²(p-b)(p-c)] / [a²bc], etc.
  22. AI = r/sin(A/2), etc.
  23. AI² = (p - a)bc/p, etc.
  24. AI² = bc·tan(B/2)·tan(C/2), etc.
  25. 1/r = 1/ra + 1/rb + 1/rc.
  26. ra + rb + rc = r + 4R.
  27. rarbrc = pS.
  28. r rarbrc = S².
  29. ra² = p(p - a)-1(p - b)(p - c), etc.
  30. la² = 4p(p - a)bc(b + c)-2, etc.
  31. la = 2bc cos(A/2) / (b + c), etc.
  32. ma² = (b² + c²)/2 - a²/4, etc.
  33. abc = 4RS.
  34. bc = 2Rha.
  35. OI² = R² - 2Rr.
  36. r + ra + rb - rc = 4Rcos(C).
  37. ra + rb = 4Rcos²(C/2).
  38. p = 4Rcos(A/2)·cos(B/2)·cos(C/2).
  39. p - a = 4Rcos(A/2)·sin(B/2)·sin(C/2).
  40. S = 2R²sin(A)·sin(B)·sin(C).
  41. r = 4Rsin(A/2)·sin(B/2)·sin(C/2).
  42. 2vers(A)vers(B)vers(C) = r²/R².
  43. cot(A/2) + cot(B/2) + cot(C/2) = cot(A/2)·cot(B/2)·cot(C/2).
  44. vers(A)vers(B)vers(C) = 8·AI·BI·CI/R3
  45. rR = abc / 4p
  46. AH = 2R·|cos(A)|
  47. AH2 = 4R2 - a2
  48. HO² = 9R² - (a² + b² + c²)
  49. (c + a)/b = cos((C - A)/2)/sin(B/2)
  50. (c - a)/b = sin((C - A)/2)/cos(B/2)
  51. p² = rarb + rbrc + rcra
  52. 1/(2Rr) = 1/(ab) + 1/(bc) + 1/(ca)

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