A9. Squares erected on a triangle.

Here is a nice problem from Coxeter and Greitzer’s classic book, Geometry Revisited, where an elegant solution is given.

Let ABC be a triangle, and construct squares externally on the sides. Let O₁ be the center of the square on AB, O₂ the center of the square on BC, and O₃ the center of the square on AC, as in the following diagram:

Prove that the segments O₁O₃ and O₂A have the same length and are perpendicular to each other.