---
book: 3
number: 19
id: "III.19"
kind: "theorem"
uses: ["[[book-3/proposition-18]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.19"
license: "CC-BY-SA-4.0"
---

# III.19

*If a straight line touch a circle*, *and from the point of contact a straight line be drawn at right angles to the tangent*, *the centre of the circle will be on the straight line so drawn*.

## Proof

For let a straight line *DE* touch the circle *ABC* at the point *C*, and from *C* let *CA* be drawn at right angles to *DE*; I say that the centre of the circle is on *AC*.

For suppose it is not, but, if possible, let *F* be the centre, and let *CF* be joined.

Since a straight line *DE* touches the circle *ABC*, and *FC* has been joined from the centre to the point of contact, *FC* is perpendicular to *DE*; [[book-3/proposition-18|III. 18]] therefore the angle *FCE* is right.

But the angle *ACE* is also right; therefore the angle *FCE* is equal to the angle *ACE*, the less to the greater: which is impossible.

Therefore *F* is not the centre of the circle *ABC*.

Similarly we can prove that neither is any other point except a point on *AC*.

Therefore etc. Q. E. D.