the midsegment of a triangle
A midsegment of a triangle is the segment joining the midpoints of two of its sides. Each triangle has three of them, and they form a smaller triangle nested inside. The midsegment has two strikingly tidy properties that make it a favourite tool: it is parallel to the third side, and it is exactly half as long.
Concretely, in triangle ABC let M be the midpoint of AB and N the midpoint of AC. Then segment MN, the midsegment, satisfies MN parallel to BC and MN = (1/2)·BC. This is really the side-splitter theorem in its cleanest case: because M and N cut the two sides in the ratio 1 : 1, the small triangle AMN is similar to ABC with scale factor 1/2 — so every length in AMN is half the corresponding length in ABC, and the matching sides are parallel. Drawing all three midsegments cuts the triangle into four smaller triangles, all congruent to one another and similar to the original at scale 1/2.
The midsegment is proportion at work in the simplest possible setting, and it generalises: the segment joining the midpoints of the two non-parallel sides of a trapezoid (the trapezoid midsegment) is parallel to the bases and equal to their average. In coordinate geometry the midsegment gives a quick proof that the three medians of a triangle meet at one point and a slick way to handle the midpoint quadrilateral (Varignon's parallelogram). It is a small theorem that opens a lot of doors.
In triangle ABC, BC = 14. M and N are the midpoints of AB and AC. The midsegment MN is parallel to BC and MN = 14/2 = 7. If you only knew MN = 7, you could deduce BC = 14.
The midsegment is parallel to the third side and exactly half its length.
Do not confuse the midsegment (joining two midpoints, parallel to the third side, half its length) with a median (joining one vertex to the midpoint of the opposite side). They are different segments; in some older texts the word 'median' was loosely used for both, which still causes confusion.