GeoProof — Basic Proportionality (Side-Splitter) explorer

DE ∥ BC ⟹ AD / DB = AE / EC

Slide the purple handle to move the parallel line. The two ratios stay equal at every height.

Basic Proportionality Theorem

Also called the side-splitter theorem or Thales' intercept theorem. If a line is drawn parallel to one side of a triangle and cuts the other two sides, it divides those two sides in the same ratio:

DE ∥ BC ⟹ AD / DB = AE / EC

Equivalently, each small piece is the same fraction of its whole side, and the cut-off segment DE is that same fraction of BC: DE / BC = AD / AB = AE / AC.

Why it works: similar triangles

Because DE ∥ BC, triangle ADE has the same angles as triangle ABC, so the two triangles are similar. Similar triangles have all their corresponding sides in one common ratio — and that shared ratio is exactly AD/AB, which forces the side splits to match.

The converse is also true

If a line cuts two sides of a triangle in equal ratios, then it must be parallel to the third side. So the equal-ratio condition and parallelism are equivalent — a handy way to prove lines parallel.

Where it's used

It is the foundation of similar-triangle geometry and scale drawings, and it justifies constructions that divide a segment into equal parts. The angle bisector theorem is a close cousin.

A bit of history

Tradition credits Thales of Miletus (c. 624–546 BCE) with using proportional intercepts to measure the height of the Great Pyramid from its shadow — an early, practical use of this idea.