Problem
GEO-B3-M03-P005 Common Point of Contact Chords
B. New Original Problem. A line \(l\) does not meet a circle \(\omega\). From a variable point \(P\in l\), two tangents to \(\omega\) are drawn, touching the circle at \(A\) and \(B\). Prove that all lines \(AB\) pass through one fixed point.
C. Hint 1. The line \(AB\) is the polar of \(P\).
D. Hint 2. Find the pole of the line \(l\) and apply La Hire.
E. Full Solution.
Let \(X\) be the pole of the line \(l\) with respect to \(\omega\). Then \(l\) is the polar of \(X\).
For every point \(P\in l\), the point \(P\) lies on the polar of \(X\). By La Hire's theorem, \(X\) lies on the polar of \(P\). But the polar of \(P\) is the chord of contact \(AB\). Hence every line \(AB\) passes through the same point \(X\).
This is the polar version of the classical fact about a moving point and its chord of contact.