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Yet another solid geometry problem involving a cube

Source: Romanian NMO 2006, Grade 8, Problem 3

April 18, 2006
geometry3D geometry

Problem Statement

Let ABCDA1B1C1D1ABCDA_1B_1C_1D_1 be a cube and PP a variable point on the side [AB][AB]. The perpendicular plane on ABAB which passes through PP intersects the line AC′AC' in QQ. Let MM and NN be the midpoints of the segments A′PA'P and BQBQ respectively. a) Prove that the lines MNMN and BC′BC' are perpendicular if and only if PP is the midpoint of ABAB. b) Find the minimal value of the angle between the lines MNMN and BC′BC'.
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