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 O

_{1}be the center of the square on AB, O_{2}the center of the square on BC, and O_{3}the center of the square on AC, as in the following diagram:Prove that the segments O

_{1}O

_{3}and O

_{2}A have the same length and are perpendicular to each other.