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Problem 840

AIME late

Geometry Difficulty 5.6 Prove it

139. Two pairs of perpendicular lines form four right triangles. Prove that the midpoints of the hypotenuses of these triangles are the vertices of a rectangle.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

139. Let angles $B$ and $D$ of quadrilateral $ABCD$ be right angles, and let $E$ and $F$ be the points of intersection of its pairs of opposite sides. Draw the line $AC$ and establish that $AC \perp EF$ . Prove that the midpoints of the hypotenuses are vertices of a quadrilateral, the sides of which are parallel to $AC$ and $EF$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.