In a triangle $ABC$, $D$ is the midpoint of $BC$. $O_1$ is the circumcenter of $ABD$ and $O_2$ is the circumcenter of $ACD$.

$M$ is the midpoint of arc $BD$ opposite from $A$.

$N$ is the midpoint of arc $CD$ opposite from $A$.

$P$ is the midpoint of arc $AB$ opposite from $D$.

$Q$ is the midpoint of arc $AC$ opposite from $D$.

$R$ on the circumcircle of $ABD$ such that $O_1R \perp AC$.

$O_1$ and $R$ are on different sides of $AC$ or its extension.

$S$ on the circumcircle of $ACD$ such that $O_2S \perp AB$.

$O_2$ and $S$ are on different sides of $AB$ or its extension.

Prove that $MN, PS, $ and $QR$ are concurrent.

## Thursday, September 10, 2009

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