Regular Heptagon’s Midpoints Circle

Abstract

This paper explores the geometry of the regular heptagon $ABCDEFG$. We start from a classical result by Thébault and Demir that six midpoints of sides and diagonals lie on a cirlce m with diameter equal to the side of the square inscribed in the circumcircle of $ABCDEFG$. Then we discover eight more midpoints of segments on $m$ and show that they are vertices of two regular heptagons inscribed in the circle $m$. Extending further this idea we show that midpoints of many other segments also lie on the circle $m$ so that it deserves the name – the midpoints circle of $ABCDEFG$. In the proofs we use the complex numbers and perform our calculations with the help of computers in Maple V.

2000 Mathematics Subject Classification. Primary 51N20, 51M04, Secondary 14A25, 14Q05