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Hong Kong Team Selection Test 1

Geometry Difficulty 8.3 Shortlist Prove it Hong Kong

Let $\Omega$ be the incircle of $\triangle ABC$ . There are two smaller circles $\omega_1$ and $\omega_2$ inside $\triangle ABC$ . The circle $\omega_1$ is tangent to $\Omega$ at $P$ , tangent to $BC$ at $D$ , and also tangent to $AB$ . The circle $\omega_2$ is tangent to $\Omega$ at $Q$ , tangent to $BC$ at $E$ , and also tangent to $AC$ . Prove that $D, E, Q, P$ are concyclic.

Solution

Let $S$ be the point on $\Omega$ different from the contact point of $\Omega$ and $BC$ such that its tangent to $\Omega$ is parallel to $BC$ . Then the tangent at $D$ to $\omega_1$ is parallel to the tangent at $S$ to $\Omega$ . By considering the homothety with centre $P$ mapping $\omega_1$ to $\Omega$ , we see that $D$ , $P$ , $S$ are collinear. Similarly, $E$ , $Q$ , $S$ are collinear. Let $X$ be a point on the tangent at $S$ to $\Omega$ as shown. Then we have
$\angle SQP = \angle PSX = \angle EDP.$
This implies $D$ , $E$ , $Q$ , $P$ are concyclic.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.