Spoiler:Let
sides of
sides of
sides of
sides of .
Then we have
and
Using Heron's formula, we have
Suppose you have a right-angled tetrahedron, i.e. a tetrahedron with a vertex where three faces meet at right angles. Let be the area of the face opposite this vertex, and the areas of the remaining faces. Show that .
Edit : I had first posted a more general problem but I'll put it on hold for the time being.
Let
together with the origin , they form a right-angled tetrahedron
we have
and their cross product is
we also know that the area of the face opposite this vertex is
the half of the modulus of the cross product so
where are the area of the lateral faces of the tetrahedron .