Posts tagged 'nine-point-circle'

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.52 Homogeneous Barycentric Coordinates

Every triangle center X(n) has homogeneous barycentric coordinates (u : v : w) normalizable to absolute weights lambda_A + lambda_B + lambda_C = 1 such that X = lambda_A A + lambda_B B + lambda_C C; for the nine-point center N = X(5), lambda_A = 1/4 + (b^2 - c^2)(a^2 - (b^2 + c^2)) / (32 K^2).