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condition for a parallelepiped to be a cube

Source: Romanian District Olympiad 2015, Grade VIII, Problem 4

September 25, 2018
Pure geometry3D geometrygeometry

Problem Statement

Consider the rectangular parallelepiped ABCDA′B′C′D′ ABCDA'B'C'D' and the point O O to be the intersection of AB′ AB' and A′B. A'B. On the edge BC, BC, pick a point N N such that the plane formed by the triangle B′AN B'AN has to be parallel to the line AC′, AC', and perpendicular to DO′. DO'. Prove, then, that this parallelepiped is a cube.
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