In a triangle, G is the centroid of triangle ADC. AE is perpendicular to FC. BD = DC and AC = 12. Find AB. G is the centroid of the triangle ADC. AE is perpendicular to FC. BD = DC
and AC = 12. Find AB.

According to the solution manual, we can let the midpoint of AC be H. D, G, and H are collinear as G is the centroid. Given that AGC is a right triangle, AG is 6, and DG is 12. How come the DG is 6 and DG is 12?


Given AG $\perp$ CG, the midpoint H is the circumcenter of AGC, which yields GH = $\frac12$AC = 6 and in turn DH = 3GH = 18 due to the centroid point G. Then, AB = 2DH = 36 since D and H are the midpoints.
