Difference between revisions of "2000 Pan African MO Problems/Problem 5"

Latest revision as of 00:10, 28 January 2023

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.

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 {{Pan African MO box
 |year=2000
-|before=First Problem
+|num-b=4
-|num-a=2
+|num-a=6
 }}
 [[Category:Geometry Problems]] {{stub}}

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

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Difference between revisions of "2000 Pan African MO Problems/Problem 5"

Latest revision as of 00:10, 28 January 2023

Solution

See also