Problem
GEO-B2-M10-P016 Equal Tangents to Different Circles
#16
★★★★☆ Level 4 of 5
Circles \(\omega_1\) and \(\omega_2\) meet at \(A,B\). Point \(P\) lies on line \(AB\). From \(P\), tangents \(PX\) to \(\omega_1\) and \(PY\) to \(\omega_2\) are drawn. Prove that \(PX=PY\).
Point \(P\) lies on the radical axis, so its powers with respect to the circles are equal.
Line \(AB\) is the radical axis, so \(\operatorname{Pow}_{\omega_1}(P)=\operatorname{Pow}_{\omega_2}(P)\). These powers are \(PX^2\) and \(PY^2\), because \(PX\) and \(PY\) are tangents. Therefore \(PX^2=PY^2\), hence \(PX=PY\).
The problem connects radical axis and tangents.