1962 IMO Problems/Problem 6
Consider an isosceles triangle. Let be the radius of its circumscribed circle and the radius of its inscribed circle. Prove that the distance between the centers of these two circles is
<geogebra>d1f93636341cbe0bc2f98c788171d8a55d94f8c8</geogebra> Instead of an isosceles triangle, let us consider an arbitrary triangle . Let have circumcenter and incenter . Extend to meet the circumcircle again at . Then extend so it meets the circumcircle again at . Consider the point where the incircle meets , and let this be point . We have ; thus, , or . Now, drawing line , we see that . Therefore, is isosceles, and . Substituting this back in, we have . Extending to meet the circumcircle at , we see that by Power of a Point. Therefore, , and we have , and we are done.
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