Given triangles ABC and DEF, find the center O and radius k of the circle of inversion such that the inverses A', B', C' of A, B, C form a triangle congruent to $\Delta DEF.$ (This is problem 5.3.6 from Geometry Revisited, by Coxeter and Greitzer).
The diagram below illustrates the construction given in the back, but I don't understand how to prove that it holds true.
Here is the description of the construction from the book:
Construct an isosceles triangle $BO_1C$ with equal angles $A + D - 90^\circ$ at B and C, and an isosceles triangle $CO_2A$ with equal angles $B + E - 90^\circ$ at C and A. Circles through C with centers $O_1, O_2$ meet again at the desired center O. The radius k is given by $$k^2 = \frac{OA \cdot OB \cdot DE}{AB}.$$
