(a) Given a point O inside an equilateral triangle △ABC. Line OG connects O with the center of mass G of the triangle and intersects the sides of the triangle, or their extensions, at points A′,B′,C′ . Prove that A′GA′O+B′GB′O+C′GC′O=3.
(b) Point G is the center of the sphere inscribed in a regular tetrahedron ABCD. Straight line OG connecting G with a point O inside the tetrahedron intersects the faces at points A′,B′,C′,D′. Prove that A′GA′O+B′GB′O+C′GC′O+D′GD′O=4. geometryratioEquilateralCentroid3D geometryspheretetrahedron