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 1 of 6: Name the circle centers by the angle they sit in

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O_A,\,O_B,\,O_C \text{: centers of the circles inscribed in angles } A,B,C

Detailed analysis

Let the circle tangent to sides $AB,AC$ have center $O_A$ , and similarly define $O_B$ , tangent to $BC,BA$ , and $O_C$ , tangent to $CA,CB$ . All three circles are congruent and pass through the common point $O$ .