Midline in Triangle

In a triangle, a midline (or a midsegment) is any of the three lines joining the midpoints of any pair of the sides of the triangle.

In a triangle, the midline joining the midpoints of two sides is parallel to the third side and half as long. Conversely, the line joining points on two sides of a triangle, parallel to its third side and half as long is a midline.

One proof is a direct consequence of Thales' Theorem. Here we offer another proof.

Proof

Let AD = BD and AE = CE. Prove that DE||BC and DE = BC/2.

Extend DE beyond E to F such that DE = EF. Since AE = CE, triangles ADE and CEF are equal, making CF||AB (or CF||BD, which is the same) because, for the transversal AC, the alternating angles DAE and ECF are equal. Also, CF = AD = BD, such that BDFC is a parallelogram. It follows that BC = DF = 2·DE which is what we set out to prove.

Conversely, let D be on AB, E on AC, DE||BC and DE = BC/2. Prove that AD = DB and AE = CE.

This is so because the condition DE||BC makes triangles ADE and ABC similar, with implied proportion,

AB/AD = AC/AE = BC/DE = 2.

It thus follows that AB is twice as long as AD so that D is the midpoint of AB; similarly, E is the midpoint of AC.


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