Problem
GEO-B3-M02-P010 Tangents at Opposite Vertices
#10
★★☆☆☆ Level 2 of 5
B. New Original Problem. Points \(A,B,C,D\) lie on one circle. The tangents to the circle at \(A\) and \(C\) meet at \(X\). Let \(Y=AB\cap CD\) and \(Z=BC\cap AD\). Prove that \(X,Y,Z\) are collinear.
Inspired by Prasolov projective geometry method
C. Hint 1. Use a hexagon with repeated vertices.
D. Hint 2. Apply Pascal to \(A,A,B,C,C,D\).
E. Full Solution.
Consider the degenerate hexagon \(A,A,B,C,C,D\) inscribed in the given circle. The side between the neighbouring points \(A,A\) is the tangent at \(A\), and the side \(C,C\) is the tangent at \(C\).
The opposite sides of this hexagon meet at \(X\), \(Y\), and \(Z\). By Pascal's theorem these three points are collinear.
This is the first essential degenerate Pascal form and prepares the module on poles and polars.