Midpoint theorem in a triangle

The segment joining midpoints of two sides is parallel to the third side and half as long.

Key formulas

D, E midpoints of AB, AC → DE ∥ BC and DE = ½ BC.

Often proves a smaller triangle similar to the original, or shows a parallelogram in a Varignon setting.

BC=10 → DE=5. If BC has gradient m, DE has the same gradient.

Problem. In △ABC, D and E are midpoints of AB and AC. If BC = 18 cm, find DE.

Solution. By the midpoint theorem, DE ∥ BC and DE = BC/2 = 9 cm.

A line through the midpoint parallel to the third side bisects the other side.

BC=14, D and E midpoints of AB and AC. DE equals:

No — the theorem is general.