*To find the centre of a given circle.*

Let *TeX Embedding failed!* be the given circle;

thus it is required to find the centre of the circle *TeX Embedding failed!*.

Let a straight line *TeX Embedding failed!* be drawn through it at random, and let it be bisected at the point *TeX Embedding failed!*;

from *TeX Embedding failed!* let *TeX Embedding failed!* be drawn at right angles to *TeX Embedding failed!* and let it be drawn through to *TeX Embedding failed!*;

let *TeX Embedding failed!* be bisected at *TeX Embedding failed!*;

I say that *TeX Embedding failed!* is the centre of the circle *TeX Embedding failed!*.

For suppose it is not, but, if possible, let *TeX Embedding failed!* be the centre,

and let *TeX Embedding failed!*, *TeX Embedding failed!*, *TeX Embedding failed!* be joined.

Then, since *TeX Embedding failed!* is equal to *TeX Embedding failed!*,

and *TeX Embedding failed!* is common,

the two sides *TeX Embedding failed!*, *TeX Embedding failed!* are equal to the two sides *TeX Embedding failed!*, *TeX Embedding failed!* respectively;

and the base *TeX Embedding failed!* is equal to the base *TeX Embedding failed!* for they are radii;

therefore the angle *TeX Embedding failed!* is equal to the angle *TeX Embedding failed!*. [Prop. 1.8]

But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right; [Def. 1.10]

therefore the angle *TeX Embedding failed!* is right.

But the angle *TeX Embedding failed!* is also right;

therefore the angle *TeX Embedding failed!* is equal to the angle *TeX Embedding failed!*, the greater to the less; which is impossible.

Therefore *TeX Embedding failed!* is not the centre of the circle *TeX Embedding failed!*.

Similarly we can prove that neither is any other point except *TeX Embedding failed!*.

Therefore the point *TeX Embedding failed!* is the centre of the circle *TeX Embedding failed!*.

[PORISM. From this it is manifest that, if in a circle a straight line cut a straight line into two equal parts and at right angles, the centre of the circle is on the cutting straight line.]

Q.E.F.