Problem
GEO-B3-M03-P015 Locus of Tangent Intersections
B. New Original Problem. A point \(P\) is fixed with respect to a circle \(\omega\). All secants through \(P\) meet \(\omega\) at \(A\) and \(B\). Let \(T\) be the intersection of the tangents at \(A\) and \(B\). Prove that all such points \(T\) lie on one line, and identify this line.
C. Hint 1. For each secant \(AB\), the point \(T\) has polar \(AB\).
D. Hint 2. Since \(P\in AB\), apply La Hire.
E. Full Solution.
For each secant \(AB\), the tangents at \(A\) and \(B\) meet at a point \(T\), whose polar is \(AB\). Since \(P\in AB\), the point \(P\) lies on the polar of \(T\).
By La Hire's theorem, \(T\) lies on the polar of \(P\). Hence all points \(T\) lie on one fixed line - the polar of \(P\) with respect to \(\omega\). Conversely, every admissible point of this line arises from some secant through \(P\), so this is the desired locus in the natural domain.
This is one of the most important formulations: the polar as the locus of tangent intersections.