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International Zhautykov Olympiad 2011 - Problem 1

Source:

January 16, 2011
geometrytrapezoidcircumcircleparallelogramrectangleperpendicular bisectorPythagorean Theorem

Problem Statement

Given is trapezoid ABCDABCD, MM and NN being the midpoints of the bases of ADAD and BCBC, respectively. a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment MNMN. b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line MNMN?
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