Let's try to render the first part of the proof a bit more legibly.
Point F is a midpoint of Line segment AB
Point E is a midpoint of Line segment AC
Draw Line segment BE
Draw Line segment FC by Construction
Point G is the point of intersection between Line segment BE and Line segment FC Intersecting Lines Postulate
Draw Line segment AG by Construction
Point D is the point of intersection between Line segment AG and Line segment BC Intersecting Lines Postulate
Point H lies on Line segment AG such that Line segment AG ≅ Line segment GH by Construction
OK, now we continue. We need to prove some parallel lines; statement 4 lets us do so.
IV Line segment FG is parallel to line segment BH and Line segment GE is parallel to line segment HC -------- Midsegment Theorem
Now that we've shown some segments parallel we extend that to collinear segments.
III Line segment GC is parallel to line segment BH and Line segment BG is parallel to line segment HC -------- Substitution
We have enough parallel lines to prove a parallelogram
I BGCH is a parallelogram -------- Properties of a Parallelogram (opposite sides are parallel)
Now we draw conclusions from that.
II Line segment BD ≅ Line segment DC -------- Properties of a Parallelogram (diagonals bisect each other)
Answer: IV III I II, second choice
