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0612 5 points lie on sphere 6th edition Round 1 p2

Source:

May 3, 2021
geometry3D geometrysphererectangle6th edition

Problem Statement

Let ABCDABCD be a rectangle of center OO in the plane α\alpha, and let V∉αV \notin\alpha be a point in space such that VO⊥αV O \perp \alpha. Let A′∈(VA)A' \in (V A), B′∈(VB)B'\in (V B), C′∈(VC)C'\in (V C), D′∈(VD)D'\in (V D) be four points, and let MM and NN be the midpoints of the segments A′C′A'C' and B′D′B'D'. .Prove that MN∥αMN \parallel \alpha if and only if V,A′,B′,C′,D′V , A', B', C', D' all lie on a sphere.
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