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Saudi Arabian Mathematical Competitions

Geometry Difficulty 4.5 AIME Prove it Saudi Arabia

Let $ABC$ be a triangle with $\angle A < \angle B \leq \angle C$ , $M$ and $N$ the midpoints of sides $CA$ and $AB$ , respectively, and $P$ and $Q$ the projections of $B$ and $C$ on the medians $CN$ and $BM$ , respectively. Prove that the quadrilateral $MNPQ$ is cyclic.

Solution

Because $\angle BPC = \angle BQC = 90^\circ$ , quadrilateral $BCQP$ is cyclic and therefore
$\angle CPQ = \angle CBQ.$
Because $M$ and $N$ are midpoints of sides $AC$ and $AB$ , segment $MN$ is parallel to side $BC$ . Hence $\angle NMQ = \angle CBQ$ .

We deduce that $\angle NMQ = \angle CPQ$ . This proves that quadrilateral $MNPQ$ is cyclic.

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