2004 Indonesia MO Problems/Problem 4
There exists 4 circles, , such that is tangent to both and , is tangent to both and , is both tangent to and , and is both tangent to and . Show that all these tangent points are located on a circle.
Let be the centers of circle and respectively. Also, let the tangent point of circles and be , let the tangent point of circles and be , let the tangent point of circles and be , and let the tangent point of circles and be . Finally, let be and respectively.
Note that are isosceles triangles, so degrees, degrees, degrees, and degrees. The sum of the angles in a line is so degrees and degrees.
That means degrees. Since is a quadrilateral, so Thus, all the tangent points of the four circles can be located in another circle.
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