IMO Shortlist 1981 problem 2
Dodao/la:
arhiva2. travnja 2012. A sphere
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
is tangent to the edges
![AB,BC,CD,DA](/media/m/b/c/4/bc4a92785cbf90193ba08ecb615b81fb.png)
of a tetrahedron
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
at the points
![E,F,G,H](/media/m/b/f/2/bf288fcefc1d79f4df24992a2dfcb86b.png)
respectively. The points
![E,F,G,H](/media/m/b/f/2/bf288fcefc1d79f4df24992a2dfcb86b.png)
are the vertices of a square. Prove that if the sphere is tangent to the edge
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
, then it is also tangent to the edge
%V0
A sphere $S$ is tangent to the edges $AB,BC,CD,DA$ of a tetrahedron $ABCD$ at the points $E,F,G,H$ respectively. The points $E,F,G,H$ are the vertices of a square. Prove that if the sphere is tangent to the edge $AC$, then it is also tangent to the edge $BD.$
Izvor: Međunarodna matematička olimpijada, shortlist 1981