Posts tagged 'incircle'

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.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).

F.29 Inradius

The inradius r = K / s = (1/2) sqrt(((b+c-a)(c+a-b)(a+b-c))/(a+b+c)) = 4 R sin(alpha/2) sin(beta/2) sin(gamma/2) takes integer values on primitive Pythagorean triples and partitions the perimeter into Conway segments x, y, z whose touchpoints T_A, T_B, T_C concur at the Gergonne point Ge = X(7).

F.32 Markowski’s Theorem

If J, K, L denote the perpendicular sagittas from the side midpoints of triangle ABC to the circumcircle, Markowski’s Theorem states J K L = (1/2) r^2 R = (R^2 - d_IO^2)^2 / (8 R).

F.48 Verisimilitude, Corner Subtriangles, and Excircle Homothety

If lines parallel to the sides of triangle ABC are drawn tangent to the incircle, the three corner subtriangles are homothetic to ABC with scale factors x/s, y/s, z/s (where x = s - a, y = s - b, z = s - c), so their inradii satisfy r_A + r_B + r_C = r.