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Taiwan IMO Selection Camp

Geometry Difficulty 6.5 National Olympiad Prove it Taiwan

Let $ABC$ be an acute triangle with $AB < AC$ . Let $\Omega$ be the circumcircle of $\triangle ABC$ . Let $B_0$ be the midpoint of $AC$ , $C_0$ be the midpoint of $AB$ , and let $\Omega_1$ be the circumcircle of $\triangle AB_0C_0$ . Let $\omega$ be a circle passing through $B_0$ and $C_0$ that is tangent to $\Omega$ at a point $X$ different from $A$ . Let $a$ be the common tangent line of $\Omega$ and $\Omega_1$ , and let $x$ be the common tangent line of $\Omega$ and $\omega$ . Prove that: the three lines $a, x$ and $B_0C_0$ are concurrent.

(2) Continuing from (1), let $D$ be the foot of the perpendicular from $A$ to $BC$ , and let $G$ be the centroid of $\triangle ABC$ . Prove that the three points $D, G, X$ are collinear.

Solution

(1) Note that $a$ is the radical axis of $\Omega$ and $\Omega_1$ , $x$ is the radical axis of $\Omega$ and $\omega$ , and $B_0C_0$ is the radical axis of $\Omega_1$ and $\omega$ . Based on the fact that the three radical axes determined by three circles are concurrent, we obtain: the three lines $a, x$ and $B_0C_0$ are concurrent.

(2) Let $O$ be the circumcenter of $\triangle ABC$ , $A_0$ be the midpoint of $BC$ , and $Q$ be the foot of the perpendicular from $A_0$ to $B_0C_0$ . Note that $\angle WAO = \angle WQO = \angle WXO = 90^\circ$ , so the five points $A, W, X, O, Q$ are concyclic. Furthermore, note that reflection about $B_0C_0$ maps $A$ to $D$ , and reflection about $OW$ maps $A$ to $X$ . Therefore,
$\angle WQD = \angle WQA = \angle WXA = \angle WAX = \angle WQX.$
Therefore the three points $Q, D, X$ are collinear.
Finally, note that the spiral similarity (homothety-rotation) centered at the centroid $G$ , with ratio $1:2$ and rotation angle $180^\circ$ , maps $\triangle ABC$ to $\triangle A_0B_0C_0$ , and simultaneously maps $AD$ to $A_0Q$ .
Hence the three points $D, G, Q$ are collinear, and thus the three points $D, G, X$ are collinear. This completes the proof.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from zh; metadata (topic, difficulty) added by this project.