Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.3 AIME, harder Prove it

7. As shown in the upper right figure, $AB$ is the diameter of semicircle $O$ . Chords $AC$ and $BD$ are drawn through $A$ and $B$ , respectively. Let $AC$ and $BD$ intersect at $E$ . Tangents to the circle are drawn through $C$ and $D$ , intersecting at point $P$ . Connect $PE$ . Prove: $PE \perp AB$ .

Solution

( Hint: Connect $A D$ )、 $BC$ intersect at $F$ , then $E$ is the orthocenter of $\triangle F A B$ , so $F E \perp A B$ , it is sufficient to prove that $P$ is on $E F$ , one way to prove that the tangent through $(\therefore D$ 's tangent lines all intersect $F E$ at the midpoint. )

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