On Th�bault's Problem 3887

Z. Cerin

Journal for geometry and graphics, Tome 15 (2011) no. 2, pp. 113-127.

Résumé

The famous Sawayama-Th�bault configuration of the triangle ABC depends on a variable point D on its sideline BC and consists of eight circles touching the lines AD and BC and its circumcircle. These circles are best considered in four pairs that are related to the four circles touching the sidelines BC, CA and AB (the incircle and the three excircles). We use analytic geometry to determine the coordinates of the centers P, Q, S, T, U, V, X, and Y of the eight Sawayama-Th�bault circles with respect to a parametrization of the triangle ABC with inradius r and cotangents f and g of the angles B/2 and C/2. The position of the point D is described by the cotangent k of half of the angle between the lines AD and BC. The coordinates of many points in this configuration are rational functions in r, f, g and k that makes most computations simple especially when done by a computer. In this approach, the proof of the original Th�bault's problem about the incenter I dividing the segment QP in the ratio k² is straightforward. Some other interesting properties of this gem of triangle geometry are explored by analytic methods.

Détail

Classification : 51N20, 51M04
Mots-clés : Triangle, line, concurrent lines, orthopole, Simson-Wallace line, locus, power