Problem
GEO-B3-M03-P010 Two Tangents and Two Chord Intersections
B. New Original Problem. Points \(A,B,C,D\) lie on a circle. The tangents at \(A\) and \(C\) meet at \(X\). Let \(Y=AB\cap CD\) and \(Z=AD\cap BC\). Prove that \(X,Y,Z\) are collinear.
C. Hint 1. Let \(T=AC\cap BD\). What is the polar of \(T\)?
D. Hint 2. Use the self-polarity of the diagonal triangle.
E. Full Solution.
Let \(T=AC\cap BD\). For the complete quadrangle \(ABCD\), the diagonal triangle is self-polar, so the polar of \(T\) is the line \(YZ\).
Since \(T,A,C\) are collinear and \(X\) is the intersection of the tangents at \(A\) and \(C\), the point \(X\) lies on the polar of \(T\). Hence \(X\in YZ\). Therefore \(X,Y,Z\) are collinear.
This is the same idea as degenerate Pascal, but in the language of polars it becomes almost immediate.