Geometry: Finding a Side Length in Triangle Using Centroid

[The Head]

The Problem is #16 in the attached picture. Essentially, I need to find the length of BC using information about congruency and the location of the centroid. I've been able to show a whole bunch of things, but nothing that gets me close to actually finding out the missing side length.

I began by drawing all of the other medians through point G, which forms triangles AGE and CGE. Some of the things I logically inferred were (note, when I use "=" below I sometimes mean congruent-- I do know the difference):
BG=GD=2GE=2ED
AE=EC
triangle AED = triangle CEG
triangle AEG = triangle CED
AG=6 (because of congruent triangles) and the distance along the median from G to the side of BC is 3 (because distance from centroid to side length is 1/3 length of median)

These congruent triangles a lot of sides and angles to be congruent, but I can't really make any progress on specific values. Please help me make some real progress with this problem!

 

[mfb]

I moved the thread to our homework section.

AG=6

That is a good start.

What do you know about the position of the centroid in a right triangle?

 

[The Head]

Ahh I can see now! It's b/3 and h/3 away from the right angle and then we can just use the Pythagorean Theorem. Thank you.