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Midpoints and parallelism

The Mid-segment Theorem is one of the most frequently used theorem in solving geometry problems. The mid-segment drawn by joining the midpoint of two sides of a triangle is always parallel to the third side, and half as long.

In the figure below, DE || BC, and DE = ½ BC

EXAMPLE

In quadrilateral ABCD, E, F, G and H are midpoints of sides AB, BC, CD and DA respectively.
Prove that line segments EG and HF bisect each other.

Proof:
Draw diagonals AC and BD.

Draw segments EF, FG, GH and HE.

In ∆ABD, since EH is a midsegment, EH || BD.

In ∆CBD, since FG is a midsegment, FG || BD.

As both EH and FG are parallel to BD, then EH || FG.

In the same way we can prove that EF || GH.

Therefore EFGH is a parallelogram, and its diagonals EG and HF bisect each other.