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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.