Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.2 AIME, harder Prove it Saudi Arabia

Pentagon $A B C D E$ is inscribed in a circle. Distances from point $E$ to lines $A B$ , $B C$ and $C D$ are equal to $a$ , $b$ and $c$ , respectively. Find the distance from point $E$ to line $A D$ .

Solution

Let $K$ , $L$ , $M$ and $N$ be the feet of perpendiculars dropped from point $E$ to lines $A B$ , $B C$ , $C D$ and $D A$ , respectively.

Points $K$ and $N$ lie on the circle with diameter $A E$ , hence $\widehat{E K N} \equiv \widehat{E A N}$ . Similarly, $\widehat{E L M} \equiv \widehat{E C M} \equiv \widehat{E A N}$ , hence
$\begin{equation*} \widehat{E K N} \equiv \widehat{E L N} . \tag{1} \end{equation*}$
In the same way we obtain
$\begin{equation*} \widehat{E N K} \equiv \widehat{E M L} . \tag{2} \end{equation*}$
From (1) and (2) it follows that $\triangle E K N \sim \triangle E L M$ , therefore
$\frac{E K}{E N}=\frac{E L}{E M}, \text{ hence } E N=\frac{E K \cdot E M}{E L}=\frac{a c}{b} .$

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