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

# III.4

*If in a circle two straight lines cut one another which are not through the centre*, *they do not bisect one another*.

## Proof

Let *ABCD* be a circle, and in it let the two straight lines *AC*, *BD*, which are not through the centre, cut one another at *E*; I say that they do not bisect one another.

For, if possible, let them bisect one another, so that *AE* is equal to *EC*, and *BE* to *ED*; let the centre of the circle *ABCD* be taken [[book-3/proposition-1|III. 1]], and let it be *F*; let *FE* be joined.

Then, since a straight line *FE* through the centre bisects a straight line *AC* not through the centre, it also cuts it at right angles; [[book-3/proposition-3|III. 3]] therefore the angle *FEA* is right.

Again, since a straight line *FE* bisects a straight line *BD*, it also cuts it at right angles; [[book-3/proposition-3|III. 3]] therefore the angle *FEB* is right.

But the angle *FEA* was also proved right; therefore the angle *FEA* is equal to the angle *FEB*, the less to the greater: which is impossible.

Therefore *AC*, *BD* do not bisect one another.

Therefore etc. Q. E. D.