MathLabs

Problem 7

The tetrahedron SABCSABC has the following property: there exist five spheres, each tangent to the edges SA,SB,SC,BC,CA,ABSA,SB,SC,BC,CA,AB, or to their extensions. (a) Prove that the tetrahedron SABCSABC is regular. (b) Prove conversely that for every regular tetrahedron five such spheres exist.
Step 5 of 6: Apply the same face argument to all four faces
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AB=BC=CA,AB=BS=AS,AS=SC=AC,SB=BC=SCAB=BC=CA,\quad AB=BS=AS,\quad AS=SC=AC,\quad SB=BC=SC
Detailed analysis

The same construction applied to faces ABSABS, ASCASC, and SBCSBC shows respectively AB=BS=ASAB=BS=AS, AS=SC=ACAS=SC=AC, and SB=BC=SCSB=BC=SC. Thus all four faces are equilateral, so SABCSABC is regular.