All faces of the tetrahedron
are acute-angled. Take a point
in the interior of the segment
, and similarly
in
in
and
in
.
a.) If
, then prove none of the closed paths
has minimal length;
b.) If
, then there are infinitely many shortest paths
, each with length
, where
.
%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 $2k=\angle BAC+\angle CAD+\angle DAB$.
Školjka