2000 USAMO Problems/Problem 5
Let be a triangle and let be a circle in its plane passing through and Suppose there exist circles such that for is externally tangent to and passes through and where for all . Prove that
Let the circumcenter of be , and let the center of be . and are externally tangent at the point , so are collinear.
is the intersection of the perpendicular bisectors of , and each of the centers lie on the perpendicular bisector of the side of the triangle that determines . It follows from that .
Since , and the perpendicular bisector of are fixed, the angle determines the position of (since lies on the perpendicular bisector). Let ; then, and together imply that .
Now (due to collinearility). Hence, we have the recursion , and so . Thus, .
implies that , and circles and are the same circle since they have the same center and go through the same two points.
Using the collinearity of certain points and the fact that is isosceles, we quickly deduce that From ASA Congruence we deduce that and are congruent triangles, and so , that is .
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