Problem of the Week 1147

Squares on a Quadrilateral

Start with any quadrilateral and erect four squares oriented outward on each of the four sides. Let A, B, C, D be the centers of these four squares, reading around the quadrilateral.

Prove that AC is perpendicular to BD, and that AC has the same length as BD.

Suggested by Dana Scott, Carnegie-Mellon University. It is known as Van Aubel's Theorem (Weisstein, Eric W. "van Aubel's Theorem." See also http://www.cut-the-knot.org/Curriculum/Geometry/SqOnQuadri.shtml#Discussion.

© Copyright 2011 Stan Wagon. Reproduced with permission.