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Romania District Olympiad 2001 - Grade VIII

Source:

March 12, 2011
geometry proposedgeometry

Problem Statement

Consider a rectangular parallelepiped ABCDA′B′C′D′ABCDA'B'C'D' in which we denote AB=a, BC=b, AA′=cAB=a,\ BC=b,\ AA'=c. Let DE⊥AC, DF⊥A′C, E∈AC, F∈A′CDE\perp AC,\ DF\perp A'C,\ E\in AC,\ F \in A'C and C′P⊥B′D′, C′Q⊥BD′, P∈B′D′, Q∈BD′C'P\perp B'D',\ C'Q\perp BD',\ P\in B'D',\ Q\in BD'. Prove that the planes (DEF)(DEF) and (C′PQ)(C'PQ) are perpendicular if and only if a2+c2=b2a^2+c^2=b^2.
Sorin Peligrad
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