Posts tagged 'incenter'

F.4 Carnot’s Theorem

In any triangle ABC, the sum of the signed perpendicular distances from the circumcenter O to the three sides BC, CA, AB equals the sum of the circumradius R and the inradius r: d_a + d_b + d_c = R + r.

F.18 Equilateral Triangle

In an equilateral triangle of side a, the incenter, centroid, circumcenter, and orthocenter coincide, with altitude h_A = (sqrt(3)/2)a = 3r and R = 2r; moreover, points A’, B’, C’ placed along the angle bisectors AI, BI, CI of any triangle ABC at distances A’I = (2r/sqrt(3)) sin(alpha/2 + pi/6) form an equilateral triangle.

F.28 Incenter

The three interior angle bisectors of triangle ABC concur at the incenter I = X(1) with homogeneous barycentric coordinates a : b : c, inradius r = K/s satisfying 1/r = 1/h_a + 1/h_b + 1/h_c, and Euler’s distance d_{IO}^2 = R(R - 2r).