Proposition 1

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.