Regular Heptagon’s Midpoints Circle - Čerin, Zvonko (Sarajevo Journal of Mathematics, 2024)
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