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 4 of 6: Make face ABC equilateral
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∣APAB∣=∣APAC∣=∣BPBC∣=∣BPBA∣=∣CPCA∣=∣CPCB∣⟹AB=BC=CA|AP_{AB}|=|AP_{AC}|=|BP_{BC}|=|BP_{BA}|=|CP_{CA}|=|CP_{CB}|\Longrightarrow AB=BC=CA
Detailed analysis

Combining the three incircle equalities with the three larger-sphere equalities makes all six displayed tangent segments equal. Each side of △ABC\triangle ABC is the sum of the two equal tangent segments at its endpoints, so AB=BC=CAAB=BC=CA and △ABC\triangle ABC is equilateral.