Construction of Thébault Circles

What Might This Be About?

Problem

Given $\Delta ABC$ and point $D$ on $BC.$

Construction of Thébault Circles - problem

Construct a circle tangent to $AD,$ $CD,$ and the circumcircle of $\Delta ABC.$

Solution

Solution proceeds in several steps:

Construction of Thébault Circles - solution

  1. Find the incenter $I$ of $\Delta ABC.$

  2. Drop a perpendicular from $I$ to the bisector of $\angle ADC.$ Mark its intersection with $BC$ - $F.$

  3. p>Erect a perpendicular to $BC$ at $F$. Mark $K$ - its intersection with the bisector of $\angle ADC.$

$K$ is the center of Thébault's circle.

For a proof, recollect Y. Sawayama's Lemma: If $F$ is the point of tangency of the Thébault circle $(K)$ and $BC$ then $IF$ passes through its point of tangency with $AD,$ say $H.$ But $FH$ is perpendicular to the bisector of $\angle ADC$ into which $(K)$ is inscribed.

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