Posts tagged 'similitude'

F.5 Centers of Similitude

Given two circles (O_1, r_1) and (O_2, r_2) separated by center distance d = |O_1 O_2| with r_2 > r_1, their common internal and external tangents concur at the internal and external centers of similitude S_IN and S_EX, which divide O_1 O_2 harmonically in the ratio r_1 : r_2.

F.16 d’Alembert’s Theorem

Given three circles of distinct radii in the plane, their six pairwise internal and external centers of similitude lie by threes on four projective lines L_{+++}, L_{–+}, L_{-+-}, and L_{+–} (the Monge–d’Alembert similitude axes).

F.44 Homothetic Ratios and Crossed Diagonals

Given two parallel uprights of heights h_1 < h_2 separated by horizontal distance d and viewed from their external center of similitude O, the crossed diagonals meet at horizontal distance x = d h_1 / (h_1 + h_2) from the first upright, while O lies at distance y - x = d h_1 / (h_2 - h_1) to its left, forming a harmonic range.

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.