Posts in category 'Triangle Centers & Classical Circles'

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.34 Nine-Point Circle

The circle Gamma_N passing through the three altitude feet H_A, H_B, H_C also passes through the three side midpoints M_A, M_B, M_C and the three Euler midpoints; its radius is r_N = R/2 and its center N = X(5) bisects segment OH on the Euler line.

F.36 Orthic Triangle

The orthic triangle H_A H_B H_C formed by the altitude feet of an acute triangle ABC is the pedal triangle of the orthocenter H = X(4), is homothetic to the tangential triangle, has H as its incenter, and solves Fagnano’s 1775 problem by minimizing the perimeter among all triangles inscribed in ABC.

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.

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).