Ref: Romanian Mathematical Olympiad — Final Round, 1994
Ref: Titu Andreescu
Let M, N, P, Q, R, S be the midpoints of the sides AB, BC, CD, DE, EF, FA of a hexagon. Prove that
if and only if MQ is perpendicular to PS.
Let a, b, c, d, e, f be the coordinates of the vertices of the hexagon. The points M, N, P, Q, R, and S have the coordinates
, , ,
, , , respectively.
Using the properties of the real product of complex numbers, (please fill in the gaps here), we have
if and only if
hence, MQ is perpendicular to PS, as claimed. QED.