The Triangle Midsegment Theorem. TMT. Sounds like fun, yes?
While not nearly as scandalous as TMZ, the TMT shares plenty of juicy morsels of gossip about the lengths of various line segments in and around the world of triangles. Specifically, it says that if you connect the midpoints of two sides of a triangle, then you've got yourself a midsegment, a magical creature that lives smack dab in the middle of the triangle it calls home. Midsegments are half the length of the side they run parallel to, they bisect the other two sides, and they fart glitter. No, wait, that's unicorns.
Here, B is the midpoint of AC, and D is the midpoint of CE. We can be sure of that because we're told that the segments are congruent on each side of both of those points. Connecting them, we get the midsegment BD. TMT says that BD || AE and that BD = 
× AE.
Sample Problem
If GI = 11, how long is FJ?
Well, we can see that GI is a midsegment, since FG ≅ GH, JI ≅ IH, and FJ || GI. This means that FJ is twice GI, so FJ = 2 × 11 = 22.
One interesting side note, or corollary if you want to be technical, is that if three parallel lines all intersect two transversals, and they create congruent segments on one of those transversals, then the segments on the other transversal are also congruent. In Triangle Land (wouldn't that be a fun theme park?), this tells us that if we have a line parallel to one side that intersects a second side at its midpoint, then it automatically intersects the third side at its midpoint as well.
(Yes, we did say three parallel lines. The third parallel line goes through the vertex opposite the first side. It's invisible, like your little brother's imaginary friend, Buster the Space Ranger.)
So, if we look at ∆FJH again, where we're given that G is the midpoint of FH and FJ || GI, then we can figure out that I is the midpoint of JH. Thanks, TMT.
Sample Problem
∆VXY has midsegment WZ, where W is the midpoint of VX. If VX = 14 and VZ = 4, how long are VW and VY?
Because WZ is a midsegment, we know it cuts VX and VY each in half, like this:
Since VX = 14 and we're slicing it in half, we know that VW = 7. And because VZ = 4, VY = 8.
But we can take this one step further. Not only does this corollary work for congruent segments, but for any proportional segments. Yep, more proportions. Please don't hate us.
Parallel lines that intersect transversals create proportional segments on those transversals.
In ∆ADE, we can set up the following proportions to find the missing lengths: 
. Let's deal with each of these separately.
Cross-multiply to get 4 × GF = 15. Divide both sides by 4 and we find GF = 3.75.
Again, cross-multiply and we get 4 × AG = 40. Divide both sides by 4 (whoa, déjà vu) and this time we have AG = 10.
Sample Problem
Find the values of a and b.
Thanks to TMT and its corollaries, we can set up proportions to find the missing lengths.
7a = 20
a = 2.857142
4b = 42
b = 10.5