Problem
GEO-B3-M01-P017 Four Circles Around a Cycle
Circles \(S_1,S_2,S_3,S_4\) are arranged so that neighboring circles meet in pairs of points \(A_i,B_i\). It is known that \(A_1,A_2,A_3,A_4\) lie on one circle. Prove that \(B_1,B_2,B_3,B_4\) also lie on one circle or one line.
Hint 1. Invert centered at one of the points \(A_i\).
Hint 2. Three circles through this center become lines.
E. Full solution. Invert centered at \(A_1\). The circles passing through \(A_1\) become lines. The circle through \(A_1,A_2,A_3,A_4\) also becomes a line. After the transformation, the statement becomes a configuration of two lines and two circles: we must prove that the four images of \(B_i\) lie on one circle or line. This follows from the usual angle criterion: the angle equality expressing cyclicity of \(A_1,A_2,A_3,A_4\) before inversion becomes an equality of angles between lines. Translating this equality back gives cyclicity or collinearity of \(B_1,B_2,B_3,B_4\).
This is a strong structure-preservation problem: one cyclicity forces another after inversion.