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rhombus wanted, DN passes through the midpoint of BE, trapezoid

Source: Dutch BxMO TST 2010 p1

August 3, 2019
geometryrhombustrapezoidmidpoint

Problem Statement

Let ABCDABCD be a trapezoid with AB//CDAB // CD, 2∣AB∣=∣CD∣2|AB| = |CD| and BD⊥BCBD \perp BC. Let MM be the midpoint of CDCD and let EE be the intersection BCBC and ADAD. Let OO be the intersection of AMAM and BDBD. Let NN be the intersection of OEOE and ABAB. (a) Prove that ABMDABMD is a rhombus. (b) Prove that the line DNDN passes through the midpoint of the line segment BEBE.
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