ABCDEF is a hexagon with vertices on a circle radius r (in that order). The three sides AB, CD, EF have length r. Show that the midpoints of BC, DE, FA form an equilateral triangle.
Solution
Use vectors. Take the origin at the center of the circle. Denote the vector OX by X. Let the midpoint of BC, DE, FA be P, Q, R respectively. Then P = (B+C)/2, Q = (D+E)/2. So the vector QP is (B+C-D-E)/2. If we rotate through 60o, then B becomes A, C becomes C-D, D becomes C, and E becomes E-F, so the vector QP becomes (A+C-D-C-E+F)/2 = (A+F-D-E)/2, which is the vector QR. Hence PQR is equilateral.
© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03