# The Midsegment of a Triangle

Submitted by Isoscel on Sat, 11/17/2007 - 18:57.

In a triangle ABC if M is the middle of AB and N is the middle of CA then MN is called the midsegment of the triangle.

MN is parallel with BC and is also half the size of BC.

Proof:

Let O be the symmetric of M in respect to N.

N is the middle of AC and the middle of MO. This means that AMCO is a parallelogram. So and .

But because M is the middle of BC we have that . So .

and A, M, B collinear means that .

and means that MBCO is a parallelogram.

MBCO is a parallelogram means that and . Because N is the middle of MO we have:

and