**Ref: Romanian Mathematical Olympiad — Final Round, 1994**

**Ref: Titu Andreescu**

**Problem:**

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.

**Proof:**

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

That is,

hence, MQ is perpendicular to PS, as claimed. **QED.**

More later,

Nalin Pithwa

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