The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular.
Let the tetrahedron be ABCD. Let G be the centroid of ABC and H the centroid of ACD. Let AM be a median in ABC and AN a median in ACD. Then AG and AH are tangents to the insphere, so they are equal. CG and CH are also tangents and hence equal. So the triangles ACG and ACH are congruent. Hence ∠AGC = ∠AHC and so ∠CGM = ∠CHN. But GM = AG/2 = AH/2 = GN, so the triangles CGM and CHN are also congruent. Hence CM = CN. Hence CB = CD. So every pair of adjacent edges is equal. Hence all the edges are equal and the tetrahedron is regular.