BPT converse — proportional ⇒ parallel

Dependencies: Basic proportionality (intercept theorem) (forward BPT), Through a point off a line, exactly one parallel exists (Playfair) (through an external point there is a unique parallel); the deepest source of uniqueness is Ruler axiom.

Statement

Let $\triangle ABC$ have $D\in AB$, $E\in AC$ satisfying

Then $DE\parallel BC$.

In other words, if two points on the two sides cut their respective sides in the same ratio, the segment joining them is automatically parallel to the third side. This is the reverse of Basic proportionality (intercept theorem) ("$DE\parallel BC\Rightarrow AD/DB=AE/EC$"): translating the "side-division ratio" back into "parallel".