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Formation of Algebraic Expressions

Prove that if...

Question

Prove that if two pairs of opposite edges at a tetrahedrun are orthogonal then third part is also orthogonal.

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Solution

Let ABCD our tetrahedron such that AB $⊥$ DC and AC $⊥$ BD
We need to prove that AD $⊥$ BC
Indeed, let $\to D A = \to a, \to D B = \to b$ and $\to D C = \to c$
Thus, $\to c \cdot (\to a - \to b) = 0$ and
$\to b \cdot (\to a - \to c) = 0$ which gives
$\to a \cdot \to c = \to b \cdot \to c$ and $\to a \cdot \to b = \to b \cdot \to c$
which gives $\to a \cdot \to c = \to a \cdot \to b$
or $\to a \cdot (\to b - \to c) = 0$ or AD $⊥$ BC.

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