Problem
GEO-B3-M01-P021 Complete Quadrilateral After Inversion
Four lines form a complete quadrilateral. One of its vertices \(P\) is chosen as the center of inversion. Prove that the circles passing through \(P\) and two neighboring vertices of the complete quadrilateral map to the sides of a certain triangle.
Hint 1. Each such circle passes through the inversion center.
Hint 2. The images of neighboring vertices lie on the images of the original lines.
E. Full solution. A circle passing through the inversion center \(P\) maps to a line. Consider the three circles through \(P\) and pairs of neighboring vertices of the complete quadrilateral. Their images are three lines. These lines meet pairwise at the images of the corresponding vertices, because inversion preserves incidence. Therefore the three image lines are the sides of a triangle whose vertices are the images of the three remaining vertices of the complete quadrilateral.
This is a preparatory problem for strong Miquel-type theorems and for turning circles into lines.