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 1 of 6: Choose the smallest sphere and its edge tangency points
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PSA,PSB,PSC,PBC,PCA,PAB=tangency points of the smallest sphereP_{SA},P_{SB},P_{SC},P_{BC},P_{CA},P_{AB}=\text{tangency points of the smallest sphere}
Detailed analysis

For part (a), choose the smallest of the five spheres and call it the smallest sphere. Denote its tangency points on the six edge lines SA,SB,SC,BC,CA,ABSA,SB,SC,BC,CA,AB by PSA,PSB,PSC,PBC,PCA,PABP_{SA},P_{SB},P_{SC},P_{BC},P_{CA},P_{AB}.