1993 USAMO Problems/Problem 2
Let be a convex quadrilateral such that diagonals and intersect at right angles, and let be their intersection. Prove that the reflections of across , , , are concyclic.
Let , , , be the foot of the altitute from point of , , , .
Note that reflection of over the 4 lines is with a scale of with center . Thus, if is cyclic, then the reflections are cyclic.
is right angle and so is . Thus, is cyclic with being the diameter of the circumcircle.
Follow that, because they inscribe the same angle.
Similarly , , .
Thus, and are supplementary and follows that, is cyclic.
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