For, with the same construction,
since, as BD is to DA, so is CE to EA,
but, as BD is to DA, so is the triangle BDE to the triangle ADE,
and, as CE is to EA, so is the triangle CDE to the triangle ADE, [VI. 1]
therefore also, as the triangle BDE is to the triangle ADE, so is the triangle CDE to the triangle ADE. [V. 11]
Therefore each of the triangles BDE, CDE has the same ratio to ADE.
Therefore the triangle BDE is equal to the triangle CDE; [V. 9]
and they are on the same base DE.
But equal triangles which are on the same base are also in the same parallels. [I. 39]
Therefore DE is parallel to BC.