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Edited by Richard Walker, Sunday, 5 May 2024, 16:49

Here's a rather neat bit of geometry. If we take a quadrilateral ABCD and join the midpoints of its sides, we get a parallelogram.

This theorem is named after Pierre Varignon (pictured, courtesy Wikipedia), a mathematician of the 17th and 18th century.

Varignon was well-connected; it seems he knew Newton and Leibnitz, for example.

Now why should the theorem be true? Well suppose we concentrate just on EH and FG and draw in the diagonal BD, see below

Now there is a theorem that says if we join the midpoints of two side of a triangle the segment so obtained is parallel to the third side and half its length. Looking at triangles ABD and DBC tells us are half the length of the diagonal and parallel to it. Hence HEFG must be a parallelogram.

We can also see that a half is not special, for example if F, G, H and E had been one-third the way along the sides they lie on, instead of one half, we would still have obtained a parallelogram.

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