perpendiculars DE and FE. Join DF; then, the angles BDE, B FE, being each a right angle, are together equal to two right angles; therefore the angles EDF, EFD are together less than two right angles; and DE, FE, produced, must meet in some point E.

Now, since the point E lies in the perpendicular DE, it is equally distant from the two points A and B (Theo. X. Bk. I.); and since the same point E lies in the perpendicular FE, it is also equally distant from the two points B and C; therefore the three distances, E A, EB, EC, are equal; hence a circumference can be described from the center E passing through the three points A, B, С.

Again, the center, lying in the perpendicular D E bisecting the chord A B, and at the same time in the perpendicular FE bisecting the chord BC (Theo. V. Cor. 2), must be at the point of their meeting, E. Therefore, since there can be but one center, but one circumference can be made to pass through three given points.

149. Cor. Two circumferences can intersect in only two points; for, if they have three points in common, they must have the same center, and must coincide.

THEOREM VII.

150. A straight line perpendicular to a radius at its termination in the circumference, is a tangent to the circle.

Let the straight line BD be per

AE

pendicular to the radius CA at its B
termination A; then will it be a tan-
gent to the circle.

D

Draw from the center C to BD any other straight line, as CE. Then, since C A is perpendicular to BD, it is shorter than the oblique

C

!

line CE (Theo. X. Bk. I.); hence the point E is without the circle. The same may be shown of any other point in the line B D, except the point A; therefore BD meets the circumference at A, and, being produced, does not cut it; hence BD is a tangent (129).

151. Cor. 1. Conversely, if a line is a tangent to a circumference, the radius drawn to the point of contact with it is perpendicular to the tangent.

For every point in BD, except A, being without the circumference, any line CE drawn from the center, C, to B D, at any point other than A, must terminate at E, without the circumference; therefore the radius CA is the shortest line that can be drawn from the center to BD; hence CA is perpendicular to the tangent B D (Theo. X. Cor. 1, Bk. I.).

152. Cor. 2. Only one tangent can be drawn through the same point in a circumference; for two lines cannot both be perpendicular to a radius at the same point.

THEOREM VIII.

153. Two parallel straight lines intercept equal arcs of the circumference.

First. When the two parallels are secants, as A B, D E. Draw the radius CH perpendicu

lar to AB; and it will also be perpendicular to DE (Theo. XVII. Cor., Bk. I.); therefore the point H will be at the same time the middle of the arc AHB and of the arc DHE (Theo. V.); therefore, the arc A H is equal to the arc HB,

H

D

E

A

B

and the arc D H is equal to the arc HE; hence A H diminished by DH is equal to HB diminished by HE; that is, the intercepted arcs AD, BE are equal.

Second. When of the two parallels, one, as AB, is a secant, and the other, as DE, is a tangent.

D

Draw the radius CH to the point of contact H. This radius will be perpendicular to the tangent DE A (Theo. VII.), and also to its parallel AB (Theo. XVII. Cor., Bk. I.). But, since CH is perpendicular to the chord A B, the point H is the middle of the arc AHB; hence the arcs AH, HB, included between the parallels A B, DE, are equal.

H

E
B

C

I

L

G

Third. When the two parallels are tangents, as DE, IL. Draw the secant AB parallel to either of the tangents; then, from what has been just shown, the arc A His equal to the arc HB, and also the arc AG is equal to the arc GB; hence the whole arc HAG is equal to the whole arc HBG.

It is further evident, since the two arcs HAG, HBG are equal, and together make up the whole circumference, that each of them is a semi-circumference.

THEOREM IX.

154. In the same circle, or in equal circles, any two angles at the center are to each other as the arcs intercepted between their sides.

Conceive the less angle to be placed on the greater; then, if the proposition be not true, the angle ACB will be to the angle ACD as the arc AB is to an arc greater or less than

i

AD. Suppose this arc to be greater, and let it be represented by AO; we shall have the angle ACB: angle ACD : : arc AB:arc AO. Conceive, now, the arc AB to be divided into equal parts, each of which is less than DO; there will be at least one point of division between D and 0; let I be that point; and join CI. The arcs AB, AI will be to each hhel loce other as two whole numbers, and, by the preceding proposition, I mean we shall have the angle ACB: angle ACI::arc AB:arc A I. Comparing these two proportions with each other, and observing that the antecedents are the same, we infer that the consequents are proportional (Theo. IX. Cor. 2, Bk. II.); hence the angle ACD: angle ACI::arc AO: arc AI. But the arc A O is greater than the arc AI; therefore, if this proportion is true, the angle A CD must be greater than the angle A CI. But it is less; hence the angle ACB cannot be to the angle ACD as the arc A B is to an arc greater than A D.

By a process of reasoning entirely similar, it may be shown that the fourth term of the proportion cannot be less than AD; therefore it must be AD; hence we have,

Angle ACB: angle ACD :: arc AB: arc A D. 155. Cor. 1. An angle having its vertex at the center of a circle is measured by the arc included between its sides.

For the angle at the center of a circle, and the arc intercepted by its sides, have such a connection, that, if the one be increased or diminished in any ratio, the other will be increased or diminished in the same ratio, we are authorized to take the one of these magnitudes as the measure of the other.

156. Cor. 2. Four right angles are measured by an entire circumference (Theo. III. Cor. 2, Bk. I.).

157. Cor. 3. Hence, a right angle is measured by a quadrant, or 90 degrees; two right angles by a semi-circumference, or 180 degrees; and four right angles by the entire circumference, or 360 degrees.

158. Scholium 1. In the comparison of angles with each

other, the arcs which serve to measure them must be described with equal radii.

159. Scholium 2. Sectors taken in the same circle, or in equal circles, are to each other as their arcs; for sectors are equal when their angles are so, and therefore are in all respects proportional to their angles.

THEOREM X.

160. An inscribed angle is measured by half the arc included between its sides.

A

The angle BCE, being exterior to the triangle A B C, is equal to the sum of the two interior angles CAB, ABC (Theo. XIX. Cor. 4, Bk. I). But the triangle BAC being isosceles, the angle CAB is equal to ABC; hence, the angle BCE is double BAC. Since BCE lies at the center, it is measured by the arc BE (Theo. IX. Cor. 1); hence BAC will be measured by half of BE. For a like reason, the angle CAD will be measured by the half of ED; hence BAC and CAD together, or BAD, will be measured by the half of BE and ED, or half B D.

Second. Suppose that the center C lies without the angle BAD. Then, drawing the diameter A E, the angle BAE will be measured by the half of BE; and the angle DAE is measured by the half of DE; hence, their differ

B

A

DE

C