Problem
GEO-B3-M01-P013 Miquel Through Inversion
Four lines in general position form four triangles. The circumcircles of three of these triangles pass through a point \(P\). Prove that the circumcircle of the fourth triangle also passes through \(P\).
Hint 1. Invert centered at \(P\).
Hint 2. Circles through \(P\) become lines.
E. Full solution. Invert centered at \(P\). The three given circles pass through the inversion center, so they become three lines. Each such line is the image of a circle passing through three vertices of one triangle of the complete quadrilateral. After inversion, the condition that three points lie on a circle through \(P\) becomes collinearity of their images. For the four original lines these are exactly the corresponding sides of the image complete quadrilateral. If three corresponding triples of images are collinear, the fourth collinearity follows from the usual Miquel theorem for four lines in reverse form. Inverting back, the fourth circle also passes through \(P\).
This problem links inversion with the Miquel point: a statement about circles becomes a linear statement.