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Pythagoras for a Tetrahedron

Age 16 to 18
Challenge Level

coordinate diagram
A natural generalisation of Pythagoras' theorem is to consider a right-angled tetrahedron with four faces, three in mutually perpendicular planes and one in the sloping plane. Then ask "what corresponds to the squares of the lengths of the sides?"

The answer must be "the squares of the areas of the faces". If these areas are $P$, $Q$ , $R$ and $S$ respectively then prove that:

$P^2+ Q^2+ R^2= S^2$
Equivalently: (area $OBC$)$^2 + $(area $OCA$)$^2 + $(area $OAB$)$^2 = $(area $ABC$)$^2$.





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