1981 IMO Problems/Problem 5

Problem

Three congruent circles have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point $O$ are collinear.

Solution

Let the triangle have vertices $A,B,C$ , and sides $a,b,c$ , respectively, and let the centers of the circles inscribed in the angles $A,B,C$ be denoted $O_A, O_B, O_C$ , respectively.

The triangles $O_A O_B O_C$ and $ABC$ are homothetic, as their corresponding sides are parallel. Furthermore, since $O_A$ lies on the bisector of angle $A$ and similar relations hold for the triangles' other corresponding points, the center of homothety is the incenter of both the triangles. Since $O$ is clearly the circumcenter of $O_A O_B O_C$ , $O$ is collinear with the incenter and circumcenter of $ABC$ , as desired.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

1981 IMO (Problems) • Resources
Preceded by Problem 4	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 6
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1981 IMO Problems/Problem 5

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