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Known cube, BP=3PD, q is DD 1. It is proved that PQ is perpendicular to the surface A 1QC 1.
Simple calculation, let the side length = 1.

pythagorean theorem

pc 1^2 = pc^2 +cc 1^2 = 2 1/8 = a 1p^2

A 1Q^2 = C 1Q^2 = 5/4

QP^2 = 3/8

PA 1^2 = A 1Q^2 + PQ^2

PC 1^2 = C 1Q^2? + PQ^2

PQ is perpendicular to A 1Q and QC 1.

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