Perpendicular

treegoner

Let ABCD be a cyclic quadilateral, O be the intersection of its diagonals AC and BD. Let M, N, P, Q be projections of O respectively on AB, BC,CD,DA. Prove that
MP is perpendicular to NQ $\Longleftrightarrow$ ABCD is a trapezoid.

**Posted:** Sun Jul 04, 2004 12:24 pm

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grobber

We use to easy-to-prove facts:

(1) $MNPQ$ is a circumscribed quadrilateral.
(2) A circumscirbed quadrilateral's diagonals are perpendicular iff one of them is a symmetry axis for the quadrilateral.

After combining these we get the solution.

**Posted:** Sun Jul 04, 2004 1:23 pm