Midline in Trapezoid

In a trapezoid, a midline (or a midsegment) is the line joining the midpoints of the sides.

In a trapezoid, the midline is parallel to the bases and its length is half their sum. Conversely, the line joining points on the two sides of a rapezoid, parallel to its bases and half as long is their sum is the midline.

Proof

Assume \(AD\) is the smaller of the two bases. Let \(AB\) and \(CD\) meet at \(E\). In \(\Delta BCE\), \(AD\parallel BC\) so that, by Thales' Theorem,

\(\displaystyle\frac{AE}{DE} = \frac{AB}{CD} = \frac{AB/2}{CD/2}.\)

If \(M'\) is the midpoint of AB and \(N'\) that of BC, then the above implies \(\displaystyle\frac{AM'}{AE} = \frac{DN'}{DE}\), from which, in turn, it follows that \(\displaystyle \frac{EM'}{AE} = \frac{EN'}{DE}\). With another reference to Thales' Theorem, M'N'||AD.

Thus there are three parallel lines and three similar triangles which supply several proportions. Of interest for us are the following:

\(\displaystyle\frac{EB}{EM'} = \frac{BC}{M'N'}\)

and

\(\displaystyle\frac{EA}{EM'} = \frac{AD}{M'N'}\),

which add up to

\(\displaystyle\frac{EB + EA}{EM'} = \frac{BC + AD}{M'N'}\)

but \(EB+EA=2EM'\) such that finally \(2M'N'=BC+AD\).

Reversing the steps yields the converse. Indeed, \(2M'N'=BC+AD\) implies \(EB+EA=2EM'\), meaning that \(M'\) is the midpoint of \(AB\). \(N'\) is similarly shown to be the midpoint of \(CD\).

The midline of a trapezoid is directly related to the midlines of triangles formed by its diagonals. In part, if \(M\) and \(N\) are the points of intersection of \(M'N'\) with \(AC\) and \(BD\), then \(M\) and \(N\) are the midpoints of \(AC\) and \(BD\), respectfully. In addition,

\(\displaystyle M'N = MN' = \frac{AD}{2}\)

and

\(\displaystyle MM' = NN' = \frac{BC}{2}\).

In a trapezoid, the line joining the midpoints \(M\) and \(N\) of the diagonals is the midline because the latter passes through these points. I mention that with a problem for young mathematicians offered at a Moscow Math Olympiad in 1957 in mind. It is considered on a separate page.

Related material Read more...
	Thales' Theorem
	Midline in Triangle
	Midline in Quadrilateral
	Bimedians in a Quadrilateral
	Midline in Similar Triangles

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