120. The Circumference or periphery of a circle is its entire bounding line; or it is a curved line, all points of which are equally distant from a point within called the center.

121. A Radius of a circle is any straight line drawn from the center to the circumference; as the line CA, CD, or CB.

122. A Diameter of a circle is any straight line drawn through the center and terminating in both directions in the circumference; as the line A B.

123. All the radii of a circle are equal; all the diameters are also equal, and each is double the radius.

arc, and the two radii drawn to the extremities of the arc; as the surface included between the arc AD, and the two radii CA, CD.

129. 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.

130. 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.

131. A straight line is inscribed in a circle when its extremities are in the circumference; as the line AB, or BC.

C

A

A

B

B

132. An Inscribed Angle is one which has its vertex in the circumference, and is formed by two chords; as the angle ABC.

133. An Inscribed Polygon is one which has the vertices of all its angles in the circumference of the circle; as the triangle ABC.

134. The circle is then said to be circumscribed about the polygon.

135. A Polygon is circumscribed about a circle when all its sides are tan- E gents to the circumference; as the polygon A B C D E F

136. The circle is then said to be

inscribed in the polygon.

THEOREM I.

F

B

A

137. Every diameter divides the circle and its circumference each into two equal parts.

Let A E B F be a circle, and A B a diameter; then the two parts AEB, AFB are equal.

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

A

F

E

B

points in the one or the other unequally distant from the center, which is contrary to the definition of the circle.

138. Hence a diameter divides the circle and its circumference into two equal parts.

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

For, let the line AB divide the circle A E B CF into two equal parts; then, if the center is not in AB, let A C be

whose chord is a diameter, is a semi-circumference, and the included segment is a semicircle.

THEOREM II.

141. 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 EGF be two equal circles, and let the arc AD be equal to E G; then will the chord AD be equal to the chord E G.

For, since the

diameters A B,

EF are equal, A

the semicircle AD B may be

applied to the

D

BE

G

F

semicircle EGF; and the curve line ADB will coincide with the curve line E G F (Theo. I.). But, by hypothesis, the arc A D is equal to the arc E G; hence the point D will fall on G; hence the chord AD is equal to the chord E G (26, Ax. 10).

Conversely, if the chord AD 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, EO G, having the three sides of the one equal to the three sides of the other, each to each, are themselves equal (Theo. XIII. Bk. I.); therefore the angle A CD is equal to the angle E O G (Theo. XIII. Sch., Bk. I.).

If now the semicircle ADB be applied to its equal EGF, with the radius AC on its cqual E O, since the angles ACD, E OG are equal, the radius CD will fall on O G, and the point D on G. Therefore the arcs AD and E G coincide with each other; hence they must be equal (26, Ax. 12).

THEOREM III.

142. 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 center, let the arc A B be greater than the arc AD; then will the chord A B be greater than the chord A D.

Draw the radii CA, CD, and CB. The two sides A C, CB in the triangle

ACB are equal to the two A C, CD in the triangle ACD, and the angle A CB is greater than the angle ACD; therefore the third side A B is greater than the third side A D (Theo. XI.

D

Bk. I.); hence the chord which subtends the greater arc is the greater.

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

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

143. 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.