Problem
GEO-B2-M03-P005 Equal Tangents and the Common Chord
#5
★★★☆☆ Level 3 of 5
Circles \(\omega_1\) and \(\omega_2\) intersect at \(A\) and \(B\). From point \(P\), outside both circles, tangents \(PT_1\) and \(PT_2\) are drawn to these circles. If \(PT_1=PT_2\), prove that \(P,A,B\) are collinear.
Equal tangents mean equal powers.
The power of point \(P\) with respect to \(\omega_1\) equals \(PT_1^2\), and with respect to \(\omega_2\) equals \(PT_2^2\). Since \(PT_1=PT_2\), the powers are equal. Thus \(P\) lies on the radical axis of the circles. They intersect at \(A\) and \(B\), so the radical axis is line \(AB\). Therefore \(P,A,B\) are collinear.
One of the main problems of the module: collinearity through equal powers.