Varignon

Varignon is a French surname. The most prominent bearer is Pierre Varignon (1654–1722), a French mathematician and engineer who contributed to geometry and mechanics and was a member of the French Academy of Sciences.

In geometry, Varignon's theorem, also called Varignon parallelogram, states that the midpoints of the four sides of any quadrilateral are the vertices of a parallelogram. If A, B, C, D are the consecutive vertices and M1, M2, M3, M4 the midpoints of AB, BC, CD, DA, then M1M2 is parallel to AC and has length |AC|/2; M2M3 is parallel to BD and has length |BD|/2; and so on. Therefore the four midpoints form a parallelogram whose area is exactly one-half the area of ABCD (for convex quadrilaterals).

The result is named after Varignon due to his publication of the idea in early 18th-century works on geometry and statics. Today, the Varignon parallelogram is a standard concept in Euclidean geometry, often used as a teaching example and a tool in proofs involving quadrilaterals.