Problem
GEO-B3-M01-P019 A Chain in an Angle
Several circles are tangent to both sides of an angle, and each is tangent to the next. Prove that the tangency points of neighboring circles lie on one line parallel to the third common tangent of any neighboring pair.
Hint 1. Circles in the same angle are homothetic with center at the vertex.
Hint 2. Neighboring circles touch on the angle bisector.
E. Full solution. All centers of the circles lie on the angle bisector. For two neighboring circles there is a homothety centered at the vertex of the angle sending the smaller circle to the larger. Their tangency point lies on the line of centers, hence on the angle bisector. Therefore all tangency points of neighboring circles lie on one line: the angle bisector. The third common tangent to two neighboring circles is also perpendicular to the line of centers in the homothetic model, and in the original angle its direction is fixed. Hence it is parallel to the corresponding tangency line.
This problem is useful as an image after inversion: chains between tangent circles often turn into chains in an angle.