---
book: 3
number: 23
id: "III.23"
kind: "theorem"
uses: ["[[book-1/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.23"
license: "CC-BY-SA-4.0"
---

# III.23

*On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side*.

## Proof

For, if possible, on the same straight line *AB* let two similar and unequal segments of circles *ACB*, *ADB* be constructed on the same side; let *ACD* be drawn through, and let *CB*, *DB* be joined.

Then, since the segment *ACB* is similar to the segment *ADB*, and similar segments of circles are those which admit equal angles, [[book-3/definitions#Definition 11|III. Def. 11]] the angle *ACB* is equal to the angle *ADB*, the exterior to the interior: which is impossible. [[book-1/proposition-16|I. 16]]

Therefore etc. Q. E. D.