2000 Pan African MO Problems/Problem 5

Let $\gamma$ be circle and let $P$ be a point outside $\gamma$. Let $PA$ and $PB$ be the tangents from $P$ to $\gamma$ (where $A, B \in \gamma$). A line passing through $P$ intersects $\gamma$ at points $Q$ and $R$. Let $S$ be a point on $\gamma$ such that $BS \parallel QR$. Prove that $SA$ bisects $QR$.

Solution

There is a projective transformation which maps $\gamma$ to a circle and that maps the midpoint of $QR$ to its center (EXPAND); therefore, we may assume without loss of generality that the midpoint of $QR$ is the center of $\gamma$. But then $B$ is the reflection of $A$ across $QR$, so that $S$ is the antipode of $A$ on $\gamma$, and we are done.

See also

2000 Pan African MO (Problems)
Preceded by
First Problem
1 2 3 4 5 6 Followed by
Problem 2
All Pan African MO Problems and Solutions

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