In the diagram below, D is the midpoint of the base of isosceles triangle ABC. Points G1 and G2 are the centroids of ABD and ACD, respectively. We know AD=8 and G1G2 =4. What is the perimeter of ABC?

From the diagram, BD = DC = G_1 G_2 = 4. Then from the Pythaogrean Theorem, AB = sqrt(4^2 + 8^2) = 4*sqrt(5), so the peimeter is AB + AC + BC = 4*sqrt(5) + 4*sqrt(5) + 8 = 8*sqrt(5) + 8.