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 5 of 6: Recognize

O

as a circumcenter

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O = \text{circumcenter of } \triangle O_AO_BO_C

Detailed analysis

The three circles are congruent and all pass through $O$ , so $O$ is equidistant from $O_A,O_B,O_C$ ; hence $O$ is the circumcenter of $\triangle O_AO_BO_C$ .