GeoProof — Triangle Midsegment explorer

each midsegment is parallel to the third side and half its length

The Triangle Midsegment

A midsegment of a triangle joins the midpoints of two of its sides. The midsegment theorem says it is always parallel to the third side and exactly half as long:

midsegment ∥ third side and midsegment = ½ × third side

Each colored midsegment here is drawn in the same color as the side it is parallel to.

The medial triangle

Joining all three midpoints forms the medial triangle. Its three sides are the three midsegments, so it has half the side lengths of the original and one-quarter of its area. It splits the big triangle into four smaller triangles that are all congruent to each other (and similar to the original at scale ½).

Why it works

It's the side-splitter theorem at the halfway mark: a line through the midpoints cuts the two sides in a 1:1 ratio, so it must be parallel to the third side, and the small triangle it forms is a half-scale copy — making the segment half the base.

Where it's used

Midsegments underlie Varignon's theorem (midpoints of a quadrilateral form a parallelogram), coordinate-geometry shortcuts, and the construction of the medial triangle and nine-point circle.

A quick check

If a side is 10 units long, the midsegment parallel to it is exactly 5 — drag a vertex and watch each midsegment stay at half its partner side.