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 6 of 6: Exhibit the five spheres for a regular tetrahedron
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rcenter=33rc,router=3rcr_{\mathrm{center}}=\frac{\sqrt3}{3}r_c,\qquad r_{\mathrm{outer}}=\sqrt3r_c
Detailed analysis

For part (b), place a regular tetrahedron with center at the origin and circumradius rcr_c. The central sphere has radius rcenter=3rc/3r_{\mathrm{center}}=\sqrt3r_c/3; a direct distance-to-edge calculation gives the same distance from the origin to all six edge lines. For each of the four faces, place a congruent outer sphere on the outward normal through the opposite face; in the same coordinates its radius is router=3rcr_{\mathrm{outer}}=\sqrt3r_c, and symmetry gives tangency to all six extended edge lines. Thus there are one central and four outer spheres.