Problem
GEO-B3-M03-P014 Tangents at Opposite Vertices
B. New Original Problem. Points \(A,B,C,D\) lie on a circle. The tangents at \(B\) and \(D\) meet at \(X\). Let \(Y=AB\cap CD\) and \(Z=AD\cap BC\). Prove that \(X,Y,Z\) are collinear.
C. Hint 1. Introduce the point \(T=BD\cap AC\).
D. Hint 2. What is the polar of \(T\) in the complete quadrangle?
E. Full Solution.
Let \(T=BD\cap AC\). For the complete quadrangle \(ABCD\), the polar of \(T\) is the line through the other two diagonal points, namely through \(Y\) and \(Z\).
Since \(T,B,D\) are collinear and \(X\) is the intersection of the tangents at \(B\) and \(D\), the point \(X\) lies on the polar of \(T\). Hence \(X\in YZ\), as required.
This resembles the previous task, but the vertex order is different; it checks flexibility of the method.