Problem
GEO-B3-M01-P011 Circle Through Two Points and Tangency
Given points \(A\), \(B\), and a line \(l\) not passing through \(A\). Construct a circle through \(A\) and \(B\) tangent to \(l\). Justify the construction using inversion.
Hint 1. Invert centered at \(A\).
Hint 2. The desired circle becomes a line through \(B^*\).
E. Full solution. Perform an arbitrary inversion centered at \(A\). Line \(l\) maps to a circle \(l^*\) through \(A\). The desired circle passes through the inversion center \(A\), so its image is a line. Since it passes through \(B\), this line must pass through \(B^*\). Tangency to \(l\) is preserved, so the image line must be tangent to circle \(l^*\). Thus draw the tangents from \(B^*\) to \(l^*\), then invert these lines back. Their images are the required circles.
The problem does not require completing the straightedge-and-compass construction in detail; the key is that tangency becomes an ordinary tangent from a point.