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Brazilian Mathematical Olympiad

Geometry Difficulty 4.8 AIME Prove it Brazil

Let $ABCDE$ be a regular pentagon such that the star $ACEBD$ has area $1$ . Let $P$ be the point of intersection of $AC$ and $BE$ and $Q$ be the point of intersection of $BD$ and $CE$ . Find the area of $APQD$ .

Solution

Let $AD$ meet $EC$ and $EB$ at $R$ and $S$ respectively. $PQDR$ is a parallelogram, so the area of $PQD$ equals the area of $RQD$ , which by its turn equals the area of $ERS$ . Therefore the area of $APQD$ equals the area of $APDRES$ , which is half of the area of the whole star.

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