Problem
GEO-B2-M10-P003 A Point on the Common Chord
#3
★★☆☆☆ Level 2 of 5
Circles \(\omega_1\) and \(\omega_2\) meet at \(A,B\). Point \(P\) lies on line \(AB\). A line through \(P\) meets \(\omega_1\) at \(X,Y\), and another line through \(P\) meets \(\omega_2\) at \(U,V\). Prove that \(PX\cdot PY=PU\cdot PV\).
Line \(AB\) is the radical axis of the two circles.
Point \(P\) lies on the radical axis, so its powers with respect to \(\omega_1\) and \(\omega_2\) are equal. By the definition of power, \(PX\cdot PY=PU\cdot PV\).
A standard link: a common chord means a radical axis.