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 2 of 6: Locate each center on an angle bisector

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In plain words

Tangency to two sides is exactly the defining property of the angle bisector: equal distance to both lines.

O_A \in \text{bisector of } \angle A,\quad O_B \in \text{bisector of } \angle B,\quad O_C \in \text{bisector of } \angle C

Detailed analysis

Since $O_A$ is equidistant from lines $AB$ and $AC$ , both tangent to its circle, $O_A$ lies on the internal bisector of $\angle A$ ; likewise $O_B$ and $O_C$ lie on the bisectors of $\angle B$ and $\angle C$ .