PROP. IX. THEOR.

IF a point be taken within a circle, from which there fall more than two equal straight lines upon the circumference, that point is the centre of the circle.

Let the point D be taken within the circle ABC, from which there fall on the circumference more than two equal straight lines, DA, DB, DC; the point D is the centre of the circle.

F

DE

For, if not, let E be the centre; join DE, and produce it to the circumference in F, G; then FG is a diameter of the circle ABC. Because in FG, the diameter of the circle ABC, there is taken the point D which is not the centre, DG will be the greatest line from it to the circumferencea, and DC greater than DB, and DB than DA. But they are likewise equal, which is impossible. Therefore E is not the centre of the circle ABC. like manner it may be demonstrated that no point but D is the centre; D therefore is the centre. Wherefore, if a point be taken, &c. Q. E. D.

B

In

Book III.

a 7. 3.

Book III..

a 9. 3.

b 5. 3.

PROP. X. THEOR.

ONE circle cannot cut another in more than two points.

If it be possible, let the circumference FAB cut the circumference DEF in more than two points, B, G, F; take the centre K of the circle ABC, and join KB, KG, KF. Because within the circle DEF there is taken the point K, from which more than two equal straight lines KB, KG, KF fall on the circumference DEF, the point K isa

E

B

D

H

K

F

C

the centre of the circle DEF. But K is also the centre of the circle ABC; therefore the same point is the centre of two circles that cut each other, which is impossible. Therefore one circumference of a circle cannot cut another in more than two points. Q. E. D.

PROP. XI. THEOR.

IF two circles touch each other internally, the straight line which joins their centres being produced will pass through the point of contact.

Let the two circles ABC, ADE touch each other internally in the point A, and let F be the centre of the circle ABC, and G the centre of the circle ADE; the straight line which joins the centres F, G, being produced, passes through the point A.

For, if not, let it fall otherwise, if possible, as FGDH, and join AF, AG. Because AG, GF are great

D

E

B

era than FA, or FH.

Take away the common part FG, and Book III. greater than the remainder GH. But

the remainder AG is AG is equal to GD; therefore GD is greater than GH; and a 20. 1. it is also less, which is impossible. Therefore the straight line which joins the points F and G cannot fall otherwise than on the point A; that is, it must pass through A. Therefore, if two circles, &c. Q. E. D.

PROP. XII. THEOR.

IF two circles touch each other externally, the straight line which joins their centres will pass through the point of contact.

Let the two circles ABC, ADE touch each other externally in the point A; and let F be the centre of the circle ABC, and G the centre of ADE. The straight line which joins the points F, G, will pass through the point of contact A. For, if not, let it pass otherwise, if possible, as FCDG, and join FA, AG. Because F is the centre of the circle ABC, AF is equal to FC;

and because G is the centre of the circle ADE, AG is equal to GD. Therefore FA, AG are equal to FC,

B

DG; wherefore the

whole FG is great

E

er than FA, AG;

but it is also lessa, which is impossible. Therefore the straight a 20. 1. line which joins the points F, G, cannot pass otherwise than through the point of contact A; that is, it passes through A. Therefore, if two circles, &c. Q. E. D.

I.

Book III.

PROP. XIII. THEOR.

ONE circle cannot touch another in more points than one, whether it touches it on the inside or outside.

For, if it be possible, let the circle EBF touch the circle ABC in more points than one, and first on the inside, in the

a 10. 11. 1. points B, D; join BD, and drawa GH, bisecting BD at right angles. Because the points B, D are in the circumference of

b 2. 3.

d 11. 3.

Ꮐ

each of the circles, the straight line BD falls within each of c Cor. 1. 3. them; and their centres are in the straight line GH, which bisects BD at right angles; therefore GH passes through the point of contact. But it does not pass through it, because the points B, D are without the straight line GH, which is absurd. Therefore one circle cannot touch another in the inside in more points than one.

Nor can two circles touch each other on the outside in more than one point. For, if it be possible, let the circle ACK touch the circle ABC in the points A, C, and join AC. Because the two points A, C are in the circumference of the circle ACK, the straight line AC which joins them will fall within the circle ACK; and the circle ACK is without the circle ABC; therefore the straight line AC is without this last circle; but, because

the points A, C are in the circumference of the circle ABC, Book III. the straight line AC must be within the same circle, which is absurd. Therefore a circle cannot touch another on the outside in more than one point; and it has been shown that a circle cannot touch another on the inside in more than one point. Therefore, one circle, &c. Q. E. D.

PROP. XIV. THEOR.

EQUAL straight lines in a circle are equally dis. tant from the centre; and those which are equally distant from the centre are equal to each other.

A

Let the straight lines AB, CD, in the circle ABDC, be equal to each other; they are equally distant from the centre. Take E the centre of the circle ABDC, and from it draw EF, EG perpendicular to AB, CD; join AE and EC. Then EF bisects ABa, and EG bisects CD. But AB is equal to CD, therefore AF is equal to CG. Because AE is equal to EC, the square of AE is equal to the square of EC. Now the of AF, FE are equal to the square of AEb, and the squares of EG, GC are equal to the square of EC; therefore the squares of AF, FE are equal to the squares of CG,

squares

E

a 3. 3.

b 47. 1.

B

GE. But the square of AF is equal to the square of CG, because AF is equal to CG; therefore the remaining square of FE is equal to the remaining square of EG; therefore the straight line EF is equal to EG; therefore AB, CD are equally distant from the centres.

Next, if the straight lines AB, CD be equally distant from the centre, that is, if FE be equal to EG, AB is equal to CD. For it may be demonstrated, as before, that the squares of EF, FA are equal to the squares of EG, GC. But the square of FE is equal to the square of EG, because FE is equal to EG; therefore the square of AF is equal to the

c 3. Def. 3.