Napoleon's and van Aubel's Theorems
Starting with any triangle ABC, construct an equilateral triangle on each side, and then connect the centers of the triangles. The points form another equilateral triangle. This result is known as Napoleon's theorem.[more]
For a quadrilateral ABCD, the similar process of constructing a square on each side and then joining the centers does not result in a square. However, the line segments joining the centers of opposite squares are always the same length and at right angles to each other. This is known as van Aubel's theorem.[less]
Snapshot 1: the triangle relationship holds even if the three points are collinear
Snapshot 3: the quadrilateral relationship holds even when the line segments do not intersect
M. Gardner, Chapters 5 and 14, Mathematical Circus, New York: Alfred A. Knopf, 1979.