Disphenoid
Disphenoid is a tetrahedron whose four faces are congruent acute-angled triangles.
Main
a) A tetrahedron is a disphenoid iff
b) A tetrahedron is a disphenoid iff its circumscribed parallelepiped is right-angled.
c) Let The squares of the lengths of sides its circumscribed parallelepiped and the bimedians are:
The circumscribed sphere has radius (the circumradius):
The volume of a disphenoid is:
Each height of disphenoid
is
the inscribed sphere has radius:
where
is the area of
Proof
a)
because in
there is no equal sides.
Let consider
but one of sides need be equal
so
b) Any tetrahedron can be assigned a parallelepiped by drawing a plane through each edge of the tetrahedron parallel to the opposite edge.
is parallelogram with equal diagonals, i.e. rectangle.
Similarly, and
are rectangles.
If is rectangle, then
Similarly, is a disphenoid.
c)
Similarly,
Similarly,
Let be the midpoint
,
be the midpoint
is the bimedian of
and
The circumscribed sphere of is the circumscribed sphere of
so it is
The volume of a disphenoid is third part of the volume of so:
The volume of a disphenoid is
where
is any height.
The inscribed sphere has radius
Therefore
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