Problem
GEO-B2-M03-P003 Equal Radii
#3
★★☆☆☆ Level 2 of 5
Two circles have equal radii and centres \(O_1\) and \(O_2\). Prove that their radical axis is the perpendicular bisector of \(O_1O_2\).
For equal radii, equality of powers becomes \(XO_1=XO_2\).
Let the common radius be \(r\). Equality of powers has the form \(XO_1^2-r^2=XO_2^2-r^2\), hence \(XO_1^2=XO_2^2\), so \(XO_1=XO_2\). The locus of points equidistant from \(O_1\) and \(O_2\) is the perpendicular bisector of \(O_1O_2\).
A good connection between the new topic and a familiar locus.