Parallel Transversal Theorem

Theorem

In the words of Euclid:

If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of the triangle be cut proportionally [so that the segments adjacent to the third side are corresponding terms in the proportion], the line joining the points of section will be parallel to the remaining side of the triangle.

(The Elements: Book $\text{VI}$: Proposition $2$)

Proof

Let $\triangle ABC$ be a triangle.

Let $DE$ be drawn parallel to the side $BC$.

We need to show that $BD : DA = CE : EA$.

Let $BE$ and $CD$ be joined.

From Triangles with Equal Base and Same Height have Equal Area the area of $\triangle BDE$ is the same as the area of $\triangle CDE$.

From Ratios of Equal Magnitudes:

$\triangle BDE : \triangle ADE = \triangle CDE : \triangle ADE$

From Areas of Triangles and Parallelograms Proportional to Base:

$\triangle BDE : \triangle ADE = BD : DA$

By the same reasoning:

$\triangle CDE : \triangle ADE = CE : EA$

From Equality of Ratios is Transitive:

$BD : DA = CE : EA$

$\Box$

Now let the sides $AB, AC$ of $\triangle ABC$ be cut proportionally, so that $BD : DA = CE : EA$.

Join $DE$.

We need to show that $DE \parallel BC$.

We use the same construction as above.

From Areas of Triangles and Parallelograms Proportional to Base we have that:

$BD : DA = \triangle BDE : \triangle ADE$

$CE : EA = \triangle CDE : \triangle ADE$

From Equality of Ratios is Transitive:

$\triangle BDE : \triangle ADE = \triangle CDE : \triangle ADE$

So from Magnitudes with Same Ratios are Equal, the area of $\triangle BDE$ is the same as the area of $\triangle CDE$.

But these triangles are on the same base $DE$.

So from Equal Sized Triangles on Same Base have Same Height, it follows that $DE$ and $BC$ are parallel.

$\blacksquare$

Also known as

The parallel transversal theorem is also known as the intercept theorem.

Also see

• Midline Theorem

Historical Note

This proof is Proposition $2$ of Book $\text{VI}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VI}$. Propositions
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): intercept theorem (parallel transversal theorem)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): parallel transversal theorem