Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME Find the answer

1. As shown in Figure $4, \triangle A B C$ has an incircle $\odot O_{1}$ that touches side $B C$ at point $D, \odot O_{2}$ is the excircle of $\triangle A B C$ inside $\angle A$ . If $O_{1} B=6, O_{1} C=3, O_{1} D=2$ , then $O_{1} O_{2}=$ $\qquad$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

$\begin{aligned} & \angle O_{1} B O_{2}=\angle O_{1} C O_{2}=90^{\circ} \\ \Rightarrow & O_{1} 、 C 、 O_{2} 、 B \text { are concyclic } \\ \Rightarrow & \angle O_{1} C D=\angle O_{1} O_{2} B \\ \Rightarrow & \triangle O_{1} D C \backsim \triangle O_{1} B O_{2} \\ \Rightarrow & O_{1} O_{2}=\frac{O_{1} B \cdot O_{1} C}{O_{1} D}=\frac{3 \times 6}{2}=9 . \end{aligned}$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.