Problem
GEO-B2-M03-P009 Two Points With Equal Tangents
#9
★★★☆☆ Level 3 of 5
Two circles intersect at \(A\) and \(B\). Points \(P\) and \(Q\) lie outside both circles. From each of the points \(P,Q\), tangent lengths to the two circles are equal. Prove that \(P,Q,A,B\) are collinear.
Prove that both \(P\) and \(Q\) lie on the radical axis of the circles.
For point \(P\), equality of tangent lengths means equality of powers with respect to the two circles, so \(P\) lies on their radical axis. Similarly, \(Q\) lies on the same radical axis. Since the circles intersect at \(A\) and \(B\), the radical axis is line \(AB\). Therefore \(P,Q,A,B\) are collinear.
The problem teaches proving collinearity of several points through one common object.