Question

The angle bisectors segments AD, BD, and CD of ΔABC intersect at point D. What is the length of segment GD ? BF = 30 BD = 34

Answer 1

This is actually very similar to a problem I solved last week. Here's how you work this:

This triangle is filled with many lines and points, but these actually have names. The points are known as vertexes, the lines are actually known as medians. This outside edges of the triangle are sides. This triangle is also special, however. The point RIGHT in the middle, point D, is known as the "centroid"

The "centroid" of a triangle is known as the center of mass, everything around it is perfectly evenly balanced from that point, and for triangles, it's the point of intersection for all medians.

Now, when dealing with the centroid of triangles, there is a rule that always applies to every median: the distance from a vertex of the triangle to the centroid is ALWAYS twice that of the distance to the midpoint, which would be on the side opposite of the chosen vertex. 

The given side length of BF is 30. Point B is our chosen vertex, and line BD is equal to 34. We need to know how much is line GD worth!
Well, given our known rule, line GD SHOULD be half the length as it's buddy line BD. So, 34/2=17

To my knowledge, this is equal to 17. 
I'm not 100% on this, because this is the first i've seen the sides and medians at different angles. This should be the same, however.

~Hope this helps!

Answer 2

I'm trying to figure it out to see if he's right.