In triangle ABC, DE is parallel to BC, with D on AB and E on AC…

The basic proportionality theorem. If $DE \parallel BC$ in triangle $ABC$, with $D$ on $AB$ and $E$ on $AC$, then $\dfrac{AD}{DB} = \dfrac{AE}{EC}$. The same fact can be written $\dfrac{AD}{AB} = \dfrac{AE}{AC}$. Triangles with the same height have areas in the ratio of their bases.