Proof: By Euclid
(related to Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels)
- For since $ABCD$ is a parallelogram, $AD$ is equal to $BC$ [Prop. 1.34].
- So, for the same (reasons), $EF$ is also equal to $BC$.
- So $AD$ is also equal to $EF$.
- And $DE$ is common.
- Thus, the whole (straight line) $AE$ is equal to the whole (straight line) $DF$.
- And $AB$ is also equal to $DC$.
- So the two (straight lines) $EA$, $AB$ are equal to the two (straight lines) $FD$, $DC$, respectively.
- And angle $FDC$ is equal to angle $EAB$, the external to the internal [Prop. 1.29].
- Thus, the base $EB$ is equal to the base $FC$, and triangle $EAB$ will be equal to triangle $DFC$ [Prop. 1.4].
- Let $DGE$ have been taken away from both.
- Thus, the remaining trapezium $ABGD$ is equal to the remaining trapezium $EGCF$.
- Let triangle $GBC$ have been added to both.
- Thus, the whole parallelogram $ABCD$ is equal to the whole parallelogram $EBCF$.
- Thus, parallelograms which are on the same base and between the same parallels are equal to one another.
- (Which is) the very thing it was required to show.
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"