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trapezoid with 2 right angles, concurrency wanted (Greece Junior 2008)

Source:

July 14, 2019
geometrytrapezoidisoscelesconcurrent

Problem Statement

Let ABCDABCD be a trapezoid with AD=a,AB=2a,BC=3aAD=a, AB=2a, BC=3a and ∠A=∠B=90o\angle A=\angle B =90 ^o. Let E,ZE,Z be the midpoints of the sides AB,CDAB ,CD respectively and II be the foot of the perpendicular from point ZZ on BCBC. Prove that : i) triangle BDZBDZ is isosceles ii) midpoint OO of EZEZ is the centroid of triangle BDZBDZ iii) lines AZAZ and DIDI intersect at a point lying on line BOBO
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