Basic proportionality theorem (BPT) — a parallel cut splits the two sides proportionally

Dependencies: AA similarity (AA similarity), Parallelogram properties (opposite sides equal, diagonals bisect each other) (Parallelogram properties (opposite sides equal, diagonals bisect each other)), converse for corresponding angles (Parallel ⇒ corresponding angles equal).

Statement

Let $\triangle ABC$ have $D\in AB$ and $E\in AC$ with $DE\parallel BC$. Then

That is: a line parallel to the third side cuts the other two sides in the same ratio.