---
book: 6
number: 29
id: "VI.29"
kind: "construction"
uses: ["[[book-6/proposition-25]]", "[[book-6/proposition-21]]", "[[book-6/proposition-26]]", "[[book-1/proposition-36]]", "[[book-1/proposition-43]]", "[[book-6/proposition-24]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:6.prop.29"
license: "CC-BY-SA-4.0"
---

# VI.29

*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*.

## Proof

Let *AB* be the given straight line, *C* the given rectilineal figure to which the figure to be applied to *AB* is required to be equal, and *D* that to which the excess is required to be similar; thus it is required to apply to the straight line *AB* a parallelogram equal to the rectilineal figure *C* and exceeding by a parallelogrammic figure similar to *D*.

Let *AB* be bisected at *E*; let there be described on *EB* the parallelogram *BF* similar and similarly situated to *D*; and let *GH* be constructed at once equal to the sum of *BF*, *C* and similar and similarly situated to *D*. [[book-6/proposition-25|VI. 25]]

Let *KH* correspond to *FL* and *KG* to *FE*.

Now, since *GH* is greater than *FB*, therefore *KH* is also greater than *FL*, and *KG* than *FE*.

Let *FL*, *FE* be produced, let *FLM* be equal to *KH*, and *FEN* to *KG*, and let *MN* be completed; therefore *MN* is both equal and similar to *GH*.

But *GH* is similar to *EL*; therefore *MN* is also similar to *EL*; [[book-6/proposition-21|VI. 21]] therefore *EL* is about the same diameter with *MN*. [[book-6/proposition-26|VI. 26]]

Let their diameter *FO* be drawn, and let the figure be described.

Since *GH* is equal to *EL*, *C*, while *GH* is equal to *MN*, therefore *MN* is also equal to *EL*, *C*.

Let *EL* be subtracted from each; therefore the remainder, the gnomon *XWV*, is equal to *C*.

Now, since *AE* is equal to *EB*, *AN* is also equal to *NB* [[book-1/proposition-36|I. 36]], that is, to *LP* [[book-1/proposition-43|I. 43]].

Let *EO* be added to each; therefore the whole *AO* is equal to the gnomon *VWX*.

But the gnomon *VWX* is equal to *C*; therefore *AO* is also equal to *C*.

Therefore to the given straight line *AB* there has been applied the parallelogram *AO* equal to the given rectilineal figure *C* and exceeding by a parallelogrammic figure *QP* which is similar to *D*, since *PQ* is also similar to *EL* [[book-6/proposition-24|VI. 24]]. Q. E. F.