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A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

### Three by One

NRICH has always had good solutions from Madras College in St Andrew's, Scotland but the solutions to this problem were truly exceptional.

 Show that the edges $AD$ and $BC$ of a tetrahedron $ABCD$ are mutually perpendicular when: $AB^2+CD^2 = AC^2+BD^2$.