Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.9 AIME Prove it Austria

Let $ABC$ be an isosceles triangle with $\overline{AC} = \overline{BC}$ and $P$ be a point of the circumcircle lying on the arc $CA$ not containing $B$ .
Let $E$ and $F$ be the orthogonal projections of the point $C$ onto the lines $AP$ and $BP$ , respectively.
Prove that $AE$ and $BF$ have the same length.
W. Janous, Innsbruck

Solution

The inscribed angle theorem implies $\angle PAC = \angle PBC$ .

Abbildung 1: Problem 4.
Therefore, the right triangles $AEC$ and $BFC$ have the same angles. Since their hypotenuses have the same length $\overline{AC} = \overline{BC}$ , they are congruent and we conclude $\overline{AE} = \overline{BF}$ .

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