Posts tagged 'circumcenter'
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.11 Circumcenter
The perpendicular bisectors of the three sides of triangle ABC concur at the circumcenter O = X(3), equidistant from A, B, C by the circumradius R = abc/(4K), with homogeneous barycentric coordinates a^2 S_A : b^2 S_B : c^2 S_C.
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.