Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.4 AIME Prove it Baltic Way

Problem:

Let $ABCD$ be a rectangle and $BC = 2 \cdot AB$ . Let $E$ be the midpoint of $BC$ and $P$ an arbitrary inner point of $AD$ . Let $F$ and $G$ be the feet of perpendiculars drawn correspondingly from $A$ to $BP$ and from $D$ to $CP$ . Prove that the points $E, F, P, G$ are concyclic.

Solution

Solution:

From rectangular triangle $BAP$ we have $BP \cdot BF = AB^{2} = BE^{2}$ . Therefore the circumference through $F$ and $P$ touching the line $BC$ between $B$ and $C$ touches it at $E$ .

Analogously, the circumference through $P$ and $G$ touching the line $BC$ between $B$ and $C$ touches it at $E$ . But there is only one circumference touching $BC$ at $E$ and passing through $P$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.