Problem 7

The tetrahedron

SABC

has the following property: there exist five spheres, each tangent to the edges

SA,SB,SC,BC,CA,AB

, or to their extensions. (a) Prove that the tetrahedron

SABC

is regular. (b) Prove conversely that for every regular tetrahedron five such spheres exist.

Step 3 of 6: Compare with the larger spheres on the three edges

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|AP_{AB}|=|BP_{AB}|,\quad |BP_{BC}|=|CP_{BC}|,\quad |CP_{CA}|=|AP_{CA}|

Detailed analysis

For edge $AB$ , the cross-section of the larger sphere tangent to $AB$ and the extensions of $CA,CB$ belongs to the same sphere that is tangent to the extension of $CS$ . Since this larger circle and the incircle share the tangent line $AB$ , their tangency point on $AB$ must be the same, giving $|AP_{AB}|=|BP_{AB}|$ . The identical argument on $BC$ and $CA$ gives the other two equalities.