All faces of the tetrahedron
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
are acute-angled. Take a point
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
in the interior of the segment
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, and similarly
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
in
![BC, Z](/media/m/0/f/d/0fdb5bdab9c2c756167fc32b1e67e9a4.png)
in
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
and
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
in
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
.
a.) If
![\angle DAB+\angle BCD\ne\angle CDA+\angle ABC](/media/m/a/4/1/a41e214d38902fb56070d370d73be4c1.png)
, then prove none of the closed paths
![XYZTX](/media/m/d/0/b/d0bdc80e0db954f12835ae22a0e49d0c.png)
has minimal length;
b.) If
![\angle DAB+\angle BCD=\angle CDA+\angle ABC](/media/m/5/d/5/5d5864f849fdd0b9624e0d279b8d219b.png)
, then there are infinitely many shortest paths
![XYZTX](/media/m/d/0/b/d0bdc80e0db954f12835ae22a0e49d0c.png)
, each with length
![2AC\sin k](/media/m/1/f/5/1f5455c88e581777af7445d91f981daa.png)
, where
![2k=\angle BAC+\angle CAD+\angle DAB](/media/m/9/8/8/98861cf6b3a393ea83d542e3c8e1ba19.png)
.
%V0
All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$.
a.) If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length;
b.) If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.