Problem
GEO-B2-M02-P022 Locus of Equal Powers
#22
★★★★★ Level 5 of 5
Two circles with centres \(O_1,O_2\) and radii \(r_1,r_2\) are given. Points \(X\) and \(Y\) have equal powers with respect to these circles. Prove that line \(XY\) is perpendicular to \(O_1O_2\), if \(X\ne Y\).
Write the power of point \(X\) as \(XO_1^2-r_1^2\) and \(XO_2^2-r_2^2\).
Equality of powers for point \(X\) gives \(XO_1^2-r_1^2=XO_2^2-r_2^2\), that is \(XO_1^2-XO_2^2=r_1^2-r_2^2\). The same equality holds for point \(Y\). Thus points \(X\) and \(Y\) have the same difference of squares of distances to \(O_1\) and \(O_2\). The locus of such points is a line perpendicular to \(O_1O_2\). Therefore \(XY\perp O_1O_2\).
This is a preview of the radical axis; the next module can return to it and name it fully.