Constructing the radical axis of two given circles

In this exercise you are given two circles with centres at \(O_1\) and \(O_2\). Your task is to construct the radical axis of these two circles. When the circles intersect the problem is quite easy, so the crux of the problem is to figure out the case when the circles do not intersect.

Hint 1. You can construct points on the radical axis by drawing another circle and constructing the radical centre of the three circles (recall the radical centre is the intersection of the radical axes of the three pairs of circles). If you draw an appropriate circle then the radical centre is very easy to construct.

Hint 2. Since the radical axis is perpendicular to \(O_1 O_2\), then another way to solve the problem is to find the point \(X\) where the radical axis intersects \(O_1 O_2\). If \(r_1\) is the radius of the first circle and \(r_2\) the radius of the second circle, then we proved in class that \((|O_1 X| + |O_2 X|)(|O_1 X| - |O_2 X|) = (r_1 + r_2)(r_1 - r_2)\). Try to use this fact to construct the length \(|O_1 X| - |O_2 X|\) using similar triangles. You may need to consider two cases, depending on whether the first circle is bigger than the second, or vice versa.

The diagram below shows the radical axis of the two circles. Try moving the centres \(O_1\), \(O_2\) and the points \(P\), \(Q\) to see how the radical axis changes. The radical axis is coloured red if the second circle is bigger than the first, and coloured green if the first is bigger than the second.