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 6 of 6: Transport

O

by the homothety and conclude

Previous step Next step

I,\ O,\ O' \text{ are collinear, where } O' = \text{circumcenter of } ABC

Detailed analysis

The homothety centered at $I$ from Step 4 sends $\triangle O_AO_BO_C$ to $\triangle ABC$ , hence sends the circumcenter $O$ of the small triangle to the circumcenter $O'$ of $\triangle ABC$ . A homothety always keeps its center, any point, and that point's image on one line, so $I$ , $O$ , and $O'$ are collinear — exactly the incenter, the point $O$ , and the circumcenter.