Posts tagged 'orthocenter'

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.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).