In the diagram, G is the centroid of Triangle ABC If [ABC]=72 then what is [GME]

Barycentric coordinates!

G = (1,1,1)

M = (1/4, 1/4, 1/2)

E = (0, 1, 1)

det[G,M,E] = det[(1,1,1), (1/4,1/4,1/2), (0,1,1)] = 1/36

==> area of triangle GME = 72*1/36 = 2

The Median divides the triangle into two smaller triangles that have the same area.

[GME] = [ABC] / 24