Posts tagged 'area'

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.13 Compare Triangles in Circles

For a family of triangles ABC inscribed in a fixed circle of radius R sharing a fixed chord AB = c (with central angle 2*phi where sin(phi) = c/(2R)), the area K decomposes into the signed sum of three central isosceles triangles: K = (1/2) c h_c + (1/2) a h_a - (1/2) b h_b.

F.38 Pedal Triangles

The pedal triangle A’B’C’ of a point P(tau_A : tau_B : tau_C) has area Delta(P) = ((R^2 - OP^2)/(4 R^2)) Delta, and its six side-projection segments satisfy Carnot’s alternating square identity AC’^2 + BA’^2 + CB’^2 = C’B^2 + A’C^2 + B’A^2.