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 2 of 6: Use the incircle cross-section in face ABC
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∣APAB∣=∣APAC∣,∣BPBC∣=∣BPBA∣,∣CPCA∣=∣CPCB∣in △ABC|AP_{AB}|=|AP_{AC}|,\quad |BP_{BC}|=|BP_{BA}|,\quad |CP_{CA}|=|CP_{CB}|\quad\text{in }\triangle ABC
Detailed analysis

Each face cuts the smallest sphere in a circle tangent to the three edge lines of that face, hence the incircle of the face. In △ABC\triangle ABC, equal tangent lengths from a vertex give ∣APAB∣=∣APAC∣|AP_{AB}|=|AP_{AC}|, ∣BPBC∣=∣BPBA∣|BP_{BC}|=|BP_{BA}|, and ∣CPCA∣=∣CPCB∣|CP_{CA}|=|CP_{CB}|.