Problem 5

Three congruent circles have a common point

O

and lie inside a given triangle. Each circle is tangent to two of the triangle's sides. Prove that the incenter and the circumcenter of the triangle, together with the point

O

, are collinear.

Step 4 of 6: Identify the homothety and its center

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\triangle O_AO_BO_C \sim \triangle ABC \text{ via a homothety centered at the incenter } I

Detailed analysis

A triangle whose three sides are respectively parallel to those of $\triangle ABC$ is the image of $\triangle ABC$ under a homothety. Since $O_A$ lies on the $A$ -bisector, and similarly for $O_B,O_C$ , the bisectors of $\triangle O_AO_BO_C$ coincide with those of $\triangle ABC$ , so the homothety is centered at their common incenter $I$ .