Proof: By Euclid
(related to Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally)
 For let $BE$ and $CD$ have been joined.
 Thus, triangle $BDE$ is equal to triangle $CDE$.
 For they are on the same base $DE$ and between the same parallels $DE$ and $BC$ [Prop. 1.38].
 And $ADE$ is some other triangle.
 And equal (magnitudes) have the same ratio to the same (magnitude) [Prop. 5.7].
 Thus, as triangle $BDE$ is to [triangle] $ADE$, so triangle $CDE$ (is) to triangle $ADE$.
 But, as triangle $BDE$ (is) to triangle $ADE$, so (is) $BD$ to $DA$.
 For, having the same height  (namely), the (straight line) drawn from $E$ perpendicular to $AB$  they are to one another as their bases [Prop. 6.1].
 So, for the same (reasons), as triangle $CDE$ (is) to $ADE$, so $CE$ (is) to $EA$.
 And, thus, as $BD$ (is) to $DA$, so $CE$ (is) to $EA$ [Prop. 5.11].
 And so, let the sides $AB$ and $AC$ of triangle $ABC$ have been cut proportionally (such that) as $BD$ (is) to $DA$, so $CE$ (is) to $EA$.
 And let $DE$ have been joined.
 I say that $DE$ is parallel to $BC$.
 For, by the same construction, since as $BD$ is to $DA$, so $CE$ (is) to $EA$, but as $BD$ (is) to $DA$, so triangle $BDE$ (is) to triangle $ADE$, and as $CE$ (is) to $EA$, so triangle $CDE$ (is) to triangle $ADE$ [Prop. 6.1], thus, also, as triangle $BDE$ (is) to triangle $ADE$, so triangle $CDE$ (is) to triangle $ADE$ [Prop. 5.11].
 Thus, triangles $BDE$ and $CDE$ each have the same ratio to $ADE$.
 Thus, triangle $BDE$ is equal to triangle $CDE$ [Prop. 5.9].
 And they are on the same base $DE$.
 And equal triangles, which are also on the same base, are also between the same parallels [Prop. 1.39].
 Thus, $DE$ is parallel to $BC$.
 Thus, if some straight line is drawn parallel to one of the sides of a triangle, then it will cut the (other) sides of the triangle proportionally.
 And if (two of) the sides of a triangle are cut proportionally, then the straight line joining the cutting (points) will be parallel to the remaining side of the triangle.
 (Which is) the very thing it was required to show.
∎
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"