Problem
GEO-B3-M01-P016 Two Circles in an Angle
Two circles are tangent to both sides of an angle with vertex \(A\). Prove that the line joining their tangency points on one side of the angle is parallel to the line joining their tangency points on the other side.
Hint 1. The centers of the circles lie on the angle bisector.
Hint 2. The circles are mapped to each other by a homothety centered at \(A\).
E. Full solution. Both circles are inscribed in the same angle, so their centers lie on the angle bisector, and the circles are homothetic with center \(A\). Let the tangency points of the first circle with the sides be \(X_1,Y_1\), and those of the second be \(X_2,Y_2\). The homothety centered at \(A\) sends the first circle to the second and maps \(X_1\) to \(X_2\), \(Y_1\) to \(Y_2\). Therefore segment \(X_1Y_1\) maps to \(X_2Y_2\), and corresponding segments under homothety are parallel. Hence \(X_1Y_1\parallel X_2Y_2\).
This problem is needed before more complicated chains: after inversion, circles in one angle often appear.