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.)

Ceva's Theorem
Law of Cosines
Law of Sines
Menelaus' Theorem
Stewart's Theorem
Van Obel Theorem
Carnot's Theorem
2·S = a·h_a = b·h_b = c·h_c.
2·S = ab sin(C) = ac·sin(B) = bc·sin(A).
S² = p(p - a)(p - b)(p - c).
h_a² = 4S²/a² = 4p(p - a)(p - b)(p - c)/a², etc.
S = rp.
r² = p^-1(p - a)(p - b)(p - c).
1/r = 1/h_a + 1/h_b + 1/h_c.
h_a = c·sin(B) = b·sin(C), etc.
cot(A/2) = (p - a)/r, etc.
S = r_a(p - a), etc.
sin²(A/2) = (p - b)(p - c) / bc, etc.
cos²(A/2) = p(p - a) / bc, etc.
tan²(A/2) = (p - b)(p - c) / p(p - a), etc.
cos²[(C-B)/2] = [(b+c)²(p-b)(p-c)] / [a²bc], etc.
AI = r/sin(A/2), etc.
AI² = (p - a)bc/p, etc.
AI² = bc·tan(B/2)·tan(C/2), etc.
1/r = 1/r_a + 1/r_b + 1/r_c.
r_a + r_b + r_c = r + 4R.
r+R=R(cos(A)+cos(B)+cos(C))
r_ar_br_c = pS.
r r_ar_br_c = S².
r_a² = p(p - a)^-1(p - b)(p - c), etc.
l_a² = 4p(p - a)bc(b + c)^-2, etc.
l_a = 2bc cos(A/2) / (b + c), etc.
m_a² = (b² + c²)/2 - a²/4, etc.
abc = 4RS.
bc = 2Rh_a.
OI² = R² - 2Rr.
r + r_a + r_b - r_c = 4Rcos(C).
r_a + r_b = 4Rcos²(C/2).
p = 4Rcos(A/2)·cos(B/2)·cos(C/2).
p - a = 4Rcos(A/2)·sin(B/2)·sin(C/2).
S = 2R²sin(A)·sin(B)·sin(C).
r = 4Rsin(A/2)·sin(B/2)·sin(C/2).
2vers(A)vers(B)vers(C) = r²/R².
cot(A/2) + cot(B/2) + cot(C/2) = cot(A/2)·cot(B/2)·cot(C/2).
vers(A)vers(B)vers(C) = 8·AI·BI·CI/R³
rR = abc / 4p
AH = 2R·|cos(A)|
AH² = 4R² - a²
HO² = 9R² - (a² + b² + c²)
(c + a)/b = cos((C - A)/2)/sin(B/2)
(c - a)/b = sin((C - A)/2)/cos(B/2)
p² = r_ar_b + r_br_c + r_cr_a
1/(2Rr) = 1/(ab) + 1/(bc) + 1/(ca)

72503872