Construct a tangent line from an external point to a circle.
Audit an external-tangent construction
Problem
A tangent construction has the correct auxiliary circle but connects \(P\) to an arbitrary point \(X\) on the original circle instead of an intersection \(T\) or \(U\). Identify the first invalid step, name the valid contact points and segments, and state the perpendicular-radius test that proves tangency.
