Problem
GEO-B3-M01-P012 Circle Through a Point and Tangent to a Circle
Given points \(A\), \(B\), and a circle \(\omega\) not passing through \(A\). Construct a circle through \(A\) and \(B\) tangent to \(\omega\).
Hint 1. Invert centered at \(A\).
Hint 2. After inversion, the problem reduces to drawing a tangent from a point to a circle.
E. Full solution. Invert centered at \(A\). The desired circle, passing through \(A\), maps to a line through \(B^*\). Circle \(\omega\), not passing through \(A\), maps to another circle \(\omega^*\). Tangency is preserved, so the image line must be tangent to \(\omega^*\). Therefore draw the tangents from \(B^*\) to \(\omega^*\), then invert them back. The images of these lines are the circles through \(A\) and \(B\) tangent to \(\omega\).
This is a standard example where inversion turns a circle problem into a line problem.