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:
![\[AB'^2 = l^2 = \frac {a^2-b^2+ c^2}{2}, AC'^2 = m^2 = \frac {a^{2}-b^{2}+c^{2}}{2},\]](http://latex.artofproblemsolving.com/0/d/0/0d05e6847ad69523e8677c862123e06b50363f44.png)
![\[AD'^2 = n^2 = \frac {-a^{2}+b^{2}+c^{2}}{2}\]](http://latex.artofproblemsolving.com/c/0/9/c0937aacef3036dbee2e78ccbf43bb765696943b.png)
The circumscribed sphere has radius (the circumradius):
The volume of a disphenoid is:
![\[V= \frac {lmn}{3} = \sqrt {\frac {(a^2+b^2-c^2)(a^2-b^2+c^2)(-a^{2}+b^{2}+c^{2})}{72}}\]](http://latex.artofproblemsolving.com/f/9/5/f959df4fb3741ac832fd1e2cdd21cd7748973ddb.png)
Each height of disphenoid is ![$h=\frac {3V}{[ABC]},$](http://latex.artofproblemsolving.com/d/0/9/d09cf378bf0ce026d4a9b758f8d1810689d631e4.png)
the inscribed sphere has radius: ![$r=\frac {3V}{4[ABC]}.$](http://latex.artofproblemsolving.com/a/b/c/abc3cab7342d16a9437af88da18850d9dca7e143.png)
