Problem
GEO-B2-M03-P012 Perpendicular to the Line of Centres
#12
★★★☆☆ Level 3 of 5
Prove that the radical axis of two nonconcentric circles is perpendicular to the line joining their centres.
Use the power formula \(XO^2-r^2\).
Let the centres be \(O_1,O_2\) and the radii \(r_1,r_2\). For a point \(X\) on the radical axis, \(XO_1^2-r_1^2=XO_2^2-r_2^2\), that is \(XO_1^2-XO_2^2=r_1^2-r_2^2\). The locus of points with constant difference of squares of distances to \(O_1\) and \(O_2\) is a line perpendicular to \(O_1O_2\). Hence the radical axis is perpendicular to the line of centres.
This proof should be discussed slowly: it explains why the radical axis is a line.