Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.4 AIME, harder Prove it

3. Let $M$ be the point where the incircle of $\triangle ABC$ touches the side $AB$ , and $T$ be any point on the side $BC$ . Prove that the incircles of $\triangle BMT$ , $\triangle MTA$ , and $\triangle ATC$ are tangent to the same line.

Solution

3. First, the corresponding tangent segments are equal, thus we have
$\begin{array}{l} \text { (see figure } 1 \text { ) we get } A M \\ =A B+A C-B C- \\ A N, (A N=A M), \\ Q R=K L-P Q-R S, \\ A R=A C-L C-R S, \\ M Q=A B-A M-B K-P Q . \end{array}$
Figure 1
Therefore, $A M+Q R=A R+M Q$ , hence quadrilateral $A M Q R$ has an inscribed circle.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.