1981 IMO Problems/Problem 5
Problem
Three congruent circles have a common point 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
are collinear.
Solution
Let the triangle have vertices , and sides
, respectively, and let the centers of the circles inscribed in the angles
be denoted
, respectively.
The triangles and
are homothetic, as their corresponding sides are parallel. Furthermore, since
lies on the bisector of angle
and similar relations hold for the triangles' other corresponding points, the center of homothety is the incenter of both the triangles. Since
is clearly the circumcenter of
,
is collinear with the incenter and circumcenter of
, as desired.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.