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all but one of specific pairs of edges of 2 tetrahedrons are perpendicular

Source: Sharygin 2011 Final 10.3

March 31, 2019

geometrytetrahedronperpendicularsolid geometry

Problem Statement

Given two tetrahedrons

A_1A_2A_3A_4

and

B_1B_2B_3B_4

. Consider six pairs of edges

A_iA_j

and

B_kB_l

, where (

i, j, k, l

) is a transposition of numbers (

1, 2, 3, 4

) (for example

A_1A_2

and

B_3B_4

). It is known that for all but one such pairs the edges are perpendicular. Prove that the edges in the remaining pair also are perpendicular.