Hence there is at least one point on the circumference of O nearer to I than A or B, and therefore within the circle I. Therefore if the circumference of O meets that of I in two points it must cut it;

That is it cannot touch it in more than one point.

THEOREM VIII.

The straight line AB which joins the points of intersection A, B, is bisected at right angles by OI, which joins the centres O and I. (Fig., Theorem VII.)

For if AB be bisected at right angles the centre of either circle must lie on the bisecting line.

[1. 19.

THEOREM IX.

If two circles touch each other they have a common. tangent at the point of contact.

If one circle touch the other internally the tangent to the outer circle is obviously the tangent to the inner.

If the circles touch externally let AB be the tangent at A to 0.

Then OAB is a right angle.

And since OA is the shortest line that can be drawn to the circumference of I, OA produced must pass through the centre I.

Therefore IA is a radius, and IAB a right angle.

Therefore AB is the tangent at A to I.

COR. If two circles touch one another, the straight line joining their centres must pass through the point of contact. For the centre of either circle must lie on the normal at the point of contact.

Arcs and Angles.

THEOREM X.

In equal circles or in the same circle, equal arcs subtend equal angles at the centre, and also equal chords.

A

B

C

D

Let O and I be equal circles, and let the arc AB be equal to the arc CD.

0

the chord AB= the chord CD.

Apply the circle O to the circle I so that the point O may be on I. The circumferences will coincide throughout.

Let the circle O be turned around its centre till A coincide with C. Then since the arc AB = arc CD,

COR. 1. It is also evident that if the arc AB were greater than the arc CD the angle AOB would be greater than CID; and if less, less.

COR. 2. If the arc AB were greater than the arc CD, the chord AB would be nearer than CD to the diameter through A, and would therefore be greater than CD; and if the arc AB were less than the arc CD, the chord AB would be less than the chord CD. [II. 4. Cor. 2.

Conversely; if AOB be greater than, equal to, or less than CID, the arc AB is greater than, equal to, or less than the arc CD; or if the chord AB be greater than, equal to, or less than the chord CD, the arc AB is greater than, equal to, or less than the arc CD.

THEOREM XI.

In equal circles or in the same circle angles at the centre are to one another as the arcs on which they stand.

K

Let the circumference of the circle O be divided into a number of equal arcs AB, BC, &c. Draw the radii. Then since the arcs are equal, the angles at the centre are likewise all equal.

Hence AC being twice as great as AB, AOC is twice as great as AOB.

And AD being three times as great as AB, AOD is three times as great as AOB.

And if AK be m times as great as AB, AOK is m times as great as AOB.

Similarly if AG be n times as great as AB, AOG is n times as great as AOB.

Therefore

And

Therefore

AK: AG =m: n.

AOK: AOG =m: n.

AK: AG AOK : AOG.

=

On this property of the circle is based the ordinary method of measuring angles. The circumference of a circle is usually divided into 360 parts. The angle subtended by one of these parts is called a degree. Instruments for the measurement of angles are furnished with an arc of a circle on which these divisions are marked, and an angle is measured by ascertaining by how many of these divisions it is subtended at the centre of the circle. If the number be m the angle is m times as great as an angle of one degree, and is called an angle of m degrees. Degrees are subdivided into minutes and seconds.

THEOREM XII.

In equal circles, or in the same circle, equal straight lines are equally distant from the centre.

Let O and I be equal circles, AB and CD equal straight lines in them, these shall be equally distant from the centres.

Place O upon I and let it be turned round till the point A coincide with the point C.

A

B

Since the chord AB= the chord CD,

the arc AB= the arc CD.

[II. 10. Cor. 2.

Therefore the point B will coincide with D, and the chord AB with the chord CD.

Therefore the perpendiculars upon these lines from the centres will coincide with and be equal to one another.

COR. Also if AB be greater than CD, the arc AB must

be greater than the arc. CD, and when the circle O is placed as above on I the point B must fall beyond D. AB is then obviously nearer to the centre than CD.

Conversely. Straight lines which are equally distant from the centre are equal; also a line which is nearer to the centre is greater than one more remote.

F

D

E

G

B

THEOREM XIII.

The angle at the centre is double of the angle at the circumference which stands upon the same arc.