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Journal for Geometry and Graphics 15 (2011), No. 2, 113--127
Copyright Heldermann Verlag 2011

On Thébault's Problem 3887

Zvonko Cerin
Dept. of Mathematics, University of Zagreb, Kopernikova 7, 10020 Zagreb, Croatia

The famous Sawayama-Thébault 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-Thébault 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 Thébault'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.

Keywords: Triangle, line, concurrent lines, orthopole, Simson-Wallace line, locus, power.

MSC: 51N20; 51M04

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