---
book: 4
number: 8
id: "IV.8"
kind: "construction"
uses: ["[[book-1/proposition-10]]", "[[book-1/proposition-31]]", "[[book-1/proposition-34]]", "[[book-3/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:4.prop.8"
license: "CC-BY-SA-4.0"
---

# IV.8

*In a given square to inscribe a circle*.

## Proof

Let *ABCD* be the given square; thus it is required to inscribe a circle in the given square *ABCD*.

Let the straight lines *AD*, *AB* be bisected at the points *E*, *F* respectively [[book-1/proposition-10|I. 10]], through *E* let *EH* be drawn parallel to either *AB* or *CD*, and through *F* let *FK* be drawn parallel to either *AD* or *BC*; [[book-1/proposition-31|I. 31]] therefore each of the figures *AK*, *KB*, *AH*, *HD*, *AG*, *GC*, *BG*, *GD* is a parallelogram, and their opposite sides are evidently equal. [[book-1/proposition-34|I. 34]]

Now, since *AD* is equal to *AB*, and *AE* is half of *AD*, and *AF* half of *AB*, therefore *AE* is equal to *AF*, so that the opposite sides are also equal; therefore *FG* is equal to *GE*.

Similarly we can prove that each of the straight lines *GH*, *GK* is equal to each of the straight lines *FG*, *GE*; therefore the four straight lines *GE*, *GF*, *GH*, *GK* are equal to one another.

Therefore the circle described with centre *G* and distance one of the straight lines *GE*, *GF*, *GH*, *GK* will pass also through the remaining points.

And it will touch the straight lines *AB*, *BC*, *CD*, *DA*, because the angles at *E*, *F*, *H*, *K* are right.

For, if the circle cuts *AB*, *BC*, *CD*, *DA*, the straight line drawn at right angles to the diameter of the circle from its extremity will fall within the circle : which was proved absurd; [[book-3/proposition-16|III. 16]] therefore the circle described with centre *G* and distance one of the straight lines *GE*, *GF*, *GH*, *GK* will not cut the straight lines *AB*, *BC*, *CD*, *DA*.

Therefore it will touch them, and will have been inscribed in the square *ABCD*.

Therefore in the given square a circle has been inscribed. Q. E. F.