---
book: 12
number: 16
id: "XII.16"
kind: "construction"
uses: ["[[book-3/proposition-16]]", "[[book-10/proposition-1]]", "[[book-3/proposition-3]]", "[[book-1/proposition-4]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:12.prop.16"
license: "CC-BY-SA-4.0"
---

# XII.16

*Given two circles about the same centre*, *to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle.*

## Proof

Let *ABCD*, *EFGH* be the two given circles about the same centre *K*; thus it is required to inscribe in the greater circle *ABCD* an equilateral polygon with an even number of sides which does not touch the circle *EFGH*.

For let the straight line *BKD* be drawn through the centre *K*, and from the point *G* let *GA* be drawn at right angles to the straight line *BD* and carried through to *C*; therefore *AC* touches the circle *EFGH*. [[book-3/proposition-16|III. 16, Por.]]

Then, bisecting the circumference *BAD*, bisecting the half of it, and doing this continually, we shall leave a circumference less than *AD*. [[book-10/proposition-1|X. 1]]

Let such be left, and let it be *LD*; from *L* let *LM* be drawn perpendicular to *BD* and carried through to *N*, and let *LD*, *DN* be joined; therefore *LD* is equal to *DN*. [[book-3/proposition-3|III. 3]], [[book-1/proposition-4|I. 4]]

Now, since *LN* is parallel to *AC*, and *AC* touches the circle *EFGH*, therefore *LN* does not touch the circle *EFGH*; therefore *LD*, *DN* are far from touching the circle *EFGH*.

If then we fit into the circle *ABCD* straight lines equal to the straight line *LD* and placed continuously, there will be inscribed in the circle *ABCD* an equilateral polygon with an even number of sides which does not touch the lesser circle *EFGH*. Q. E. F.