Posts tagged 'heron'

F.3 Biquadratic Forms in Heron’s Formula

Under the linear substitution X = a/2, W = (b + c)/2, and Y = (c - b)/2, Heron’s area formula reduces to the biquadratic product K = sqrt((W^2 - X^2)(X^2 - Y^2)).

F.29 Inradius

The inradius r = K / s = (1/2) sqrt(((b+c-a)(c+a-b)(a+b-c))/(a+b+c)) = 4 R sin(alpha/2) sin(beta/2) sin(gamma/2) takes integer values on primitive Pythagorean triples and partitions the perimeter into Conway segments x, y, z whose touchpoints T_A, T_B, T_C concur at the Gergonne point Ge = X(7).

F.30 Isosceles Triangles

An isosceles triangle of given semiperimeter s and area K has half-base x = c/2 governed by the cubic 2 s x^3 - s^2 x^2 + K^2 = 0, which under x = y + s/6 reduces to Cardano’s depressed cubic y^3 - (s^2/12) y - (s^3/54 - K^2/(2s)) = 0.

F.50 Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem

Given the three altitudes h_A, h_B, h_C of triangle ABC with reciprocal semiperimeter H = (h_A^{-1} + h_B^{-1} + h_C^{-1})/2, Mitchell’s reciprocal Heron formula states K^{-1} = 4 sqrt(H (H - h_A^{-1})(H - h_B^{-1})(H - h_C^{-1})), and a triangle is right-angled at C if and only if h_A^{-2} + h_B^{-2} = h_C^{-2} (the Symphonic Theorem).