Short Answer 
The proof demonstrates key properties of a parallelogram, specifically that opposite sides are equal, diagonals bisect each other, and opposite angles are equal. By examining the intersection point O of the diagonals, it concludes that both segments of the diagonals bisect each other, confirming that EG bisects HF and vice versa.
Step 1: Identify Properties of the Parallelogram
In a parallelogram like EFGH, there are several key properties that can be utilized in the proof. These include:
- Opposite sides are equal and parallel, i.e., EF = GH and EH = FG.
- Diagonals bisect each other, meaning that the point where the diagonals intersect divides each diagonal into two equal parts.
- The opposite angles are equal, which may play a role in reasoning about triangle congruences later on.
Step 2: Establish Points of Intersection and Relationships
Let the diagonals of the parallelogram EFGH intersect at point O. By the properties of a parallelogram, this intersection divides both diagonals into two equal segments. Thus:
- EO = OG and FO = OH, indicating that diagonal EG bisects HF.
- Similarly, since diagonals bisect each other, HF also bisects EG at point O.
Step 3: Conclude the Proof
With the information established from the properties of the parallelogram and the relationships at the intersection point O, we can conclude the proof. Since both segments of the diagonals are equal, we can state that:
- EG bisects HF, meaning O is the midpoint of HF.
- HF bisects EG, confirming that O is also the midpoint of EG.
This shows the desired relationship and completes the proof of the statement that EG bisects HF and vice versa.