Problem
GEO-B3-M03-P022 Locus of Cross-Intersections
B. New Original Problem. A point \(P\) is fixed, and two variable secants through it meet a circle \(\omega\) at \(A,B\) and \(C,D\). Prove that for every choice of the secants, the points \(AC\cap BD\) and \(AD\cap BC\) lie on one fixed line. Identify this line.
C. Hint 1. Consider the complete quadrangle \(ABCD\).
D. Hint 2. The two cross-intersections form the polar of the diagonal point \(P\).
E. Full Solution.
Let \(Q=AC\cap BD\) and \(R=AD\cap BC\). Since \(A,B,C,D\) lie on the circle and \(P=AB\cap CD\), the points \(P,Q,R\) are the diagonal points of the complete quadrangle \(ABCD\).
By self-polarity of the diagonal triangle, the polar of \(P\) is the line \(QR\). The circle and the point \(P\) are fixed, so the polar of \(P\) is fixed. Therefore, for every choice of the two secants, \(Q\) and \(R\) lie on the same fixed line - the polar of \(P\).
A strong task showing that a variable complete configuration has a fixed polar axis.