PROPOSITION XVIII.-THEOREM.

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

Let BAD be an inscribed angle, whose sides include the arc BD; then the angle BAD is measured by half of the arc BD.

First. Suppose the centre of the circle C to lie within the angle BAD. Draw the diameter A E, and the radii CB, CD.

B

Α

E

The angle BCE, being exterior to the triangle ABC, is equal to the sum of the two interior angles CAB, ABC (Prop. XXVII. Bk. I.). But the triangle BAC being isosceles, the angle CAB is equal to A BC; hence, the angle BCE is double BA C. Since BCE lies at the centre, it is measured by the arc BE (Prop. XVII. Sch. 1); hence B A C 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 B AD, will be measured by the half of B E and ED, or half B D.

Second. Suppose that the centre 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 difference, BAD, will be measured by the half of BE minus the half of ED, or by the half of B D.

A

B

D E

Hence every inscribed angle is measured by the half of the arc included between its sides.

201. Cor. 1. All the angles, BAC, BDC, inscribed in the same segment, are equal; because they are all measured by the half of the same arc, BOC.

202. Cor. 2. Every angle, BAD, inscribed in a semicircle, is a right angle; because it is measured by half the semi-circumference, BOD; B that is, by the fourth part of the whole circumference.

203. Cor. 3. Every angle, BAC, inscribed in a segment greater than a semicircle, is an acute angle; for it is measured by the half of the arc BOC, less than a semi-circumference.

And every angle, BOC, inscribed in a segment less than a semicircle, is an obtuse angle; for it is measured by half of the arc BAC, greater than a semi-circumference.

204. Cor. 4. The opposite angles, A and D, of an inscribed quadrilateral, ABD C, are together equal to two right angles; for the angle BAC is measured by half the arc BDC, and the angle BDC is measured by half the arc B AC;

B

A

B

D

D

hence the two angles BA C, BD C, taken together, are measured by half the circumference; hence their sum is equal to two right angles.

PROPOSITION XIX. - THEOREM.

A C

205. The angle formed by the intersection of two chords is measured by half the sum of the two intercepted arcs. Let the two chords A B, C D intersect each other at the point E; then will the angle DE B, or its equal, AEC, be measured by half the sum of the two arcs DB and A C.

Draw A F parallel to DC; then will the arc FD be equal to the arc AC (Prop. XII.), and the an

E

B

D

gle FAB equal to the angle DEB (Prop. XXII. Bk. I.). But the angle FAB is measured by half the arc FDB (Prop. XVIII.); that is, by half the arc D B, plus half the arc FD. Hence, since FD is equal to A C, the angle DEB, or its equal angle A EC, is measured by half the sum of the intercepted arcs D B and AC

PROPOSITION XX.-THEOREM.

D

206. The angle formed by a tangent and a chord is measured by half the intercepted arc. Let the tangent BE form, with the chord A C, the angle BAC; then BAC is measured by half the arc AM C.

From A, the point of contact, draw the diameter AD. The angle BAD is a right angle (Prop. X.), and is measured by half of the semi-circumference AMD

M

B

-E

A

(Prop. XVIII.); and the angle DAC is measured by half the arc DC; hence the sum of the angles BAD, DAC, or BAC, is measured by the half of AM D, plus the half of DC; or by half the whole arc AMD C.

In like manner, it may be shown that the angle CAE is measured by half the intercepted arc A C.

PROPOSITION XXI.-THEOREM.

207. The angle formed by two secants is measured by half the difference of the two intercepted arcs.

Let AB, AC be two secants forming the angle BAC; then will that angle be measured by half the difference of the two arcs BEC and D F.

B

A

D F

E

C

Draw D E parallel to AC; then will the arc EC be equal to the arc D F (Prop. XII.); and the angle BDE be equal to the angle BAC (Prop. XXII. Bk. I.). But the angle BDE is measured by half the arc BE (Prop. XVIII.); hence the equal angle B A C is also measured by half the arc BE; that is, by half the difference of the arcs BEC and EC, or, since EC is equal to D F, by half the difference of the intercepted arcs BEC and D F.

PROPOSITION XXII.-THEOREM.

A

208. The angle formed by a secant and a tangent is measured by half the difference of the two intercepted arcs. Let the secant AB form, with the tangent A C, the angle BAC; then BAC is measured by half the difference of the two arcs BEF and FD.

Draw DE parallel to A C; then will the arc E F be equal to the arc DF (Prop. XII.), and the angle BDE be equal to the angle BAC.

B

D

F

E

But the angle BDE is measured by half of the arc BE (Prop. XVIII.); hence the equal angle BAC is also measured by half the arc BE; that is, by half the difference of the arcs BEF and EF, or, since EF is equal to DF, by half the difference of the intercepted arcs BEF and DF.

BOOK IV.

PROPORTIONS, AREAS, AND SIMILARITY OF FIGURES.

DEFINITIONS.

209. The AREA of a figure is its quantity of surface, and is expressed by the number of times which the surface contains some other area assumed as a unit of measure.

Figures have equal areas, when they contain the same unit of measure an equal number of times.

210. SIMILAR FIGURES are such as have the angles of the one equal to those of the other, each to each, and the sides containing the equal angles proportional.

211. EQUIVALENT FIGURES are such as have equal areas. Figures may be equivalent which are not similar. Thus a circle may be equivalent to a square, and a triangle to a rectangle.

212. EQUAL FIGURES are such as, when applied the one to the other, coincide throughout (Art. 34, Ax. 14). Thus circles having equal radii are equal; and triangles having the three sides of the one equal to the three sides of the other, each to each, are also equal.

Equal figures are always similar; but similar figures may be very unequal.

213. In different circles, SIMILAR ARCS, SEGMENTS, or SECTORS are such as correspond to equal angles at the centres of the circles.