Posts tagged 'inradius'

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.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.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.37 Euler’s Inradius Parabola

For a fixed circumradius R, Euler’s relation d^2 = R^2 - 2 r R defines a downward-opening parabola r(d) = R/2 - d^2/(2R) = (R - d)(R + d)/(2R) with vertex (0, R/2) at the equilateral triangle and roots at the degenerate limits d = +/- R.

F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction

Given the three interior angles (alpha, beta, gamma) of a triangle normalized to unit area K = 1 (so s = 1/r), the Conway intouch segments x = s - a = r cot(alpha/2), y = s - b = r cot(beta/2), and z = s - c = r cot(gamma/2) determine the side lengths a = y + z, b = z + x, and c = x + y = 2 s - (a + b).