# Midline Theorem

## Theorem

The midline of a triangle is parallel to the third side of that triangle and half its length.

## Proof

Let $\triangle ABC$ be a triangle.

Let $DE$ be the midline of $\triangle ABC$ through $AB$ and $AC$.

Extend $DE$ to $DF$ so $DE = EF$.

As $E$ is the midpoint of $AC$, the diagonals of the quadrilateral $ADCF$ bisect each other.

From Quadrilateral with Bisecting Diagonals is Parallelogram, $ADCF$ is a parallelogram.

By definition of a parallelogram, $AB \parallel CF$.

From Opposite Sides and Angles of Parallelogram are Equal, $AD = CF$.

But $AD = DB$ as $D$ is the midpoint of $AB$.

So $DB = CF$ and $DB \parallel CF$.

From Quadrilateral is Parallelogram iff One Pair of Opposite Sides is Equal and Parallel, $BCFD$ is a parallelogram.

Thus also by Quadrilateral is Parallelogram iff One Pair of Opposite Sides is Equal and Parallel $DF = BC$ and $DF \parallel BC$.

As $DE = EF$, $DE$ is the midpoint of $DF$ and so $DE = \dfrac 1 2 DF$.

Thus $DE = \dfrac 1 2 BC$ and $DE \parallel BC$.

Hence the result.

$\blacksquare$

## Also known as

The **Midline Theorem** is also known as the **midpoint theorem** or **mid-point theorem**.

## Sources

- 1968: M.N. Aref and William Wernick:
*Problems & Solutions in Euclidean Geometry*... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.26$: Corollary $1$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**midpoint theorem** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**triangle**: $(4)$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**mid-point theorem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**triangle**: $(4)$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**midpoint theorem**