Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME, harder Prove it United States

Problem:

There are circles $\omega_{1}$ and $\omega_{2}$ . They intersect in two points, one of which is the point $A$ . $B$ lies on $\omega_{1}$ such that $A B$ is tangent to $\omega_{2}$ . The tangent to $\omega_{1}$ at $B$ intersects $\omega_{2}$ at $C$ and $D$ , where $D$ is the closer to $B$ . $A D$ intersects $\omega_{1}$ again at $E$ . If $B D=3$ and $C D=13$ , find $E B / E D$ .

Solution

Solution:

> [diagram]

By power of a point, $B A=\sqrt{B D \cdot B C}=4 \sqrt{3}$ . Also, $D E B \sim D B A$ , so $E B / E D=B A / B D=4 \sqrt{3} / 3$ .

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