Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram

Theorem

 * To a given straight line to apply a parallelogram equal to a given rectilineal figure and exceeding by a parallelogrammic figure similar to a given one.

Construction
Let $AB$ be the given straight line, $C$ the given rectilineal figure to which the figure to be applied to $AB$ is to be equal, and $\Box D$ that to which the excess is required to be similar.

We need to apply to $AB$ a parallelogram equal to the area of $C$ exceeding it by a parallelogrammic figure similar to $\Box D$.


 * Euclid-VI-29.png

Let $AB$ be bisected at $E$.

Describe on $EB$ the parallelogram $\Box BF$ similar and similarly situated to $\Box D$.

From Construction of Figure Similar to One and Equal to Another, let $\Box GH$ be constructed at once equal to $\Box BF + C$ and similarly situated to $D$.

Let $FL$ and $FE$ be produced to $M$ and $N$ so that $FLM = KH$ and $FEN = KG$, and complete the parallelogram $\Box MN$, whose diameter is $FO$.

Construct the parallelogram $\Box AO$.

Then $\Box AO$ is the required parallelogram.