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Show that two lines are perpendicular

Source: Argentina TST 2011, Problem 3

August 31, 2014
geometrytrapezoidcircumcircleArgentinaTST

Problem Statement

Let ABCDABCD be a trapezoid with bases BC∥ADBC \parallel AD, where AD>BCAD > BC, and non-parallel legs ABAB and CDCD. Let MM be the intersection of ACAC and BDBD. Let Γ1\Gamma_1 be a circumference that passes through MM and is tangent to ADAD at point AA; let Γ2\Gamma_2 be a circumference that passes through MM and is tangent to ADAD at point DD. Let SS be the intersection of the lines ABAB and CDCD, XX the intersection of Γ1\Gamma_1 with the line ASAS, YY the intesection of Γ2\Gamma_2 with the line DSDS, and OO the circumcenter of triangle ASDASD. Show that SO⊥XYSO \perp XY.
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