ar, and de to all havia de amenities of the an as the surface boltbed between dear AD, and the two CA. CO.

151. A TANGENT to a circle is a straight line which, how far so ever produced, meets the circumference in but one point; as the line CD. The point of meeting is called the POINT OF CONTACT; as the point M.

162. Two circumferences TOUCH each other, when they have a point of contact without cutting one another; thus two circumferences touch each other at the point A, and two at the point B.

163. A STRAIGHT LINE is INSCRIBED in a circle when its extremities are in the circumference;

B

B

as the line AB, or BC.

A

164. An INSCRIBED ANGLE is one which has its vertex in the circumference, and is formed by two chords; as the

.. ABC.

165. An INSCRIBED POLYGON is one which has the vertices of all its angles in the circumference of the circle; as the triangle ABC.

C

B

166. The circle is then said to be CIRCUMSCRIBED about

168. The circle is then said to be INSCRIBED in the

polygon.

PROPOSITION I.—THEOREM.

169. Every diameter divides the circle and its circum

ference each into two equal parts.

Let AEBF be a circle, and AB a diameter; then the two parts AEB, AFB are equal.

A

F

E

B

For, if the figure A E B be applied to AFB, their common base A B retaining its position, the curve line AEB must fall exactly on the curve line AFB; otherwise there would be points in the one or the other unequally distant from the centre, which is contrary to the definition of the circle (Art. 152). Hence a diameter divides the circle and its circumference into two equal parts.

170. Cor. 1. Conversely, a straight line dividing the circle into two equal parts is a diameter.

Fir, let the ze AB divide the cirle AEBCF into two equal parts: them. if the centre is not in AB, let AC be drawn through it, which is therefore a diameter, and conse- A quently divides the circle into two equal parts: Lence the surface AFC is equal to the surface AFC B. a part to the whole, which is impossible.

171. Cor. 2. The are of a circle, whose chord is a diameter, is a semi-circumference, and the included segment is a semicircle.

172. A straight line cannot meet the circumference of a circle in more than two points.

For, if a straight line could meet the circumference ABD, in three points, A, B, D, join each of these points with the centre, C; then,

since the straight lines CA, CB,

CD are radii, they are equal (Art.

B

155); hence, three equal straight

lines can be drawn from the same point to the same straight line, which is impossible (Prop. XIV. Cor. 2, Bk. I.).

PROPOSITION III.-THEOREM.

173. In the same circle, or in equal circles, equal arcs are subtended by equal chords; and, conversely, equal chords subtend equal arcs.

Let ADB and EGP be two equal circles, and let the are AD be equal to EG; then will the chord AD be to the chord E G.

and the curve line ADB will coincide with the curve

line EGF (Prop. I.). But, by hypothesis, the arc AD is equal to the arc EG; hence the point D will fall on G; hence the chord AD is equal to the chord EG (Art. 34, Ax. 11).

Conversely, if the chord A D is equal to the chord E G, the arcs A D, E G will be equal.

For, if the radii CD, OG are drawn, the triangles ACD, EOG, having the three sides of the one equal to the three sides of the other, each to each, are themselves equal (Prop. XVIII. Bk. I.); therefore the angle A CD is equal to the angle E O G (Prop. XVIII. Sch., Bk. I.).

If now the semicircle A D B be applied to its equal EGF, with the radius AC on its equal EO, since the angles A CD, EO G are equal, the radius CD will fall on OG, and the point D on G. Therefore the arcs AD and EG coincide with each other; hence they must be equal (Art. 34, Ax. 14).

174. In the same circle, or in equal circles, a greater arc is subtended by a greater chord; and, conversely, the greater chord subtends the greater arc.

In the circle of which C is the centre, let the arc A B be greater than the arc A D; then will the chord AB be greater than the chord A D.

Draw the radii CA, CD, and C B. The two sides A

CB in the triangle ACB are equal
to the two A C, CD in the triangle
ACD, and the angle ACB is greater
than the angle ACD; therefore the A
third side A B is greater than the
third side AD (Prop. XVI. Bk. I.);
hence the chord which subtends the
greater arc is the greater.

Conversely, if the chord A B be greater than the chord AD, the arc A B will be greater than the arc A D.

For the triangles ACB, ACD have two sides, AC, CB, in the one, equal to two sides, AC, CD, in the other, while the side A B is greater than the side AD; therefore the angle ACB is greater than the angle A CD (Prop. XVII. Bk. I.); hence the arc A B is greater than the arc A D.

175. Scholium. The arcs here treated of are each less than the semi-circumference. If they were greater, the contrary would be true; in which case, as the arcs increased, the chords would diminish, and conversely.

PROPOSITION V.-THEOREM.

176. In the same circle, or in equal circles, radii which make equal angles at the centre intercept equal arcs on the circumference; and, conversely, if the intercepted arcs are equal, the angles made by the radii are also equal.

Let ACB and DCE be equal angles made by radii at the centre of equal circles; then will the intercepted arcs A B and DE be also equal.

A

B D

F

ince the angles ACB, D CE are equal, the applied to the other; and since their sides,