Posts in category 'Triangle Centers & Classical Lines'

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.6 Centroid

The three medians AM_A, BM_B, CM_C joining the vertices of triangle ABC to the midpoints of the opposite sides concur at the centroid G = X(2), which divides each median in a 2 : 1 ratio and has homogeneous barycentric coordinates 1 : 1 : 1.

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.19 Equal Incircles Theorem

If a fan of rays from apex P to a baseline cuts triangle ABC into consecutive subtriangles whose incircles have equal radii r_1 = r_2 = … = r_n = r, then the ray angles gamma_n with the altitude PR (of height h) follow the hyperbolic progression tan(gamma_n) = sinh(a + n*b) with tanh(b/2) = r/(h - r), so overlapping pairs of subtriangles also have equal incircles.

F.20 Extangent

The external common tangents to the incircle and circumcircle (or excircles) bound the extangents triangle, while Euler’s relation d_{IO}^2 = R^2 - 2rR has derivative d(d_{IO})/dR = (R - r)/sqrt(R^2 - 2rR).

F.21 Fermat Point

For any triangle ABC whose largest interior angle is less than 120 degrees, erecting external equilateral triangles BCX_A, CAY_B, ABZ_C on its sides yields concurrent segments AX_A, BY_B, CZ_C that intersect at the Fermat-Torricelli point F = X(13), where <AFB = <BFC = <CFA = 120 degrees and FA + FB + FC is minimized.

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).