Problem
GEO-B3-M03-P020 A Tangent Point on a Diagonal Line
B. New Original Problem. Points \(A,B,C,D\) lie on a circle. The tangents at \(A\) and \(D\) meet at \(S\). Let \(P=AB\cap CD\) and \(Q=AC\cap BD\). Prove that \(P,Q,S\) are collinear.
C. Hint 1. Introduce the third diagonal point \(R=AD\cap BC\).
D. Hint 2. The line \(PQ\) is the polar of \(R\), and \(S\) also lies on this polar.
E. Full Solution.
Let \(R=AD\cap BC\). In the complete quadrangle \(ABCD\), the polar of \(R\) is the line \(PQ\).
On the other hand, \(R\) lies on the chord \(AD\), and the tangents at \(A\) and \(D\) meet at \(S\). Hence, by the secant-and-tangents fact, \(S\) lies on the polar of \(R\). Therefore \(S\in PQ\), so \(P,Q,S\) are collinear.
This is an important pattern for high-level problems: introduce the missing diagonal point.