ARCS INTERCEPTED BY ANGLES.

220. An INSCRIBED ANGLE is one whose sides are chords or secants, and whose vertex is on the circumference. An angle is said to be inscribed in an arc, when its vertex is on the arc and its sides extend to or through the ends of the arc. In such a case the arc is said to contain the angle. Thus, the angle AEI is inscribed in the arc AEI, and the arc AEI contains the angle AEI.

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An angle is said to stand upon the arc intercepted between its sides. Thus, the angle AEI stands upon the arc AOI.

221. Corollary. The arc in which an angle is inscribed, and the arc intercepted between its sides, compose the whole circumference.

222. Theorem.-An inscribed angle is measured by half of the intercepted arc.

This demonstration also presents three cases. The center of the circle may be on one of the sides of the angle, or it may be inside, or it may be outside of the angle.

1st. One side of the angle, as AB, may be a diameter.

Make the diameter DE, parallel to BC, the other side of the angle. Then the angle B is equal to its alternate angle BOD (125), which is measured by the arc BD (207). This arc is equal to

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CE (219), and also to EA (197). Therefore, the arc

BD is equal to the half of AC, and the inscribed angle B is measured by half of its intercepted arc.

2d. The center of the circle may be within the angle. From the vertex B extend a diameter to the opposite side of the circumference at D.

As just proved, the angle ABD is measured by half of the arc AD, and the angle DCB by half of the arc DC. Therefore, the sum of the two angles, or ABC, is measured by half of the sum

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of the two arcs, or half of the arc ADC.

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3d. The center of the circle may be outside of the angle. Extend a diameter from the vertex as before. The angle ABC is equal to ABD diminished by DBC, and is, therefore, measured by half of the arc DA diminished by half of DC; that is, by the half of AC.

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223. Corollary.-When an inscribed angle and an angle at the center have the same intercepted arc, the inscribed angle is half of the angle at the center.

224. Corollary.-All angles inscribed in the same arc are equal, for they have the same measure.

225. Corollary.-Every angle inscribed in a semicircumference is a right angle. If the arc is less than a semi-circumference, the angle is obtuse. If the arc is greater, the angle is acute.

226. Theorem.-The angle formed by a tangent and a chord is measured by half the intercepted arc.

The angle CEI, formed by the tangent AC and the chord EI, is measured by half

the intercepted arc IDE.

Through I, make the chord IO parallel to the tangent AC.

The angle CEI is equal to its alternate EIO (125), which is measured by half the arc OME (222), which is equal to the arc

IDE (219). Therefore, the angle CEI is measured by half the arc IDE.

The sum of the angles AEI and CEI is two right angles, and is therefore measured by half the whole circumference (207). Hence, the angle AEI is equal to two right angles diminished by the angle CEI, and is measured by half the whole circumference diminished by half the arc IDE; that is, by half the arc IOME.

Thus it is proved that each of the angles formed at E, is measured by half the arc intercepted between its sides.

227. This theorem may be demonstrated very elegantly by the method of limits (200).

228. Theorem.-Every angle whose vertex is within the circumference, is measured by half the sum of the arcs intercepted between its sides and its sides produced.

Thus, the angle OAE is measured by half the sum of the arcs OE and IU.

To be demonstrated by the

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student, using the previous theorems (219 and 222).

229. Theorem.-Every angle whose vertex is outside of a circumference, and whose sides are either tangent or secant, is measured by half the difference of the intercepted arcs.

Thus, the angle ACF is measured by half the difference of the arcs AF and AB; the angle FCG, by half the difference of the arcs FG and BI; and the angle ACE, by half the difference of the arcs AFGE and ABIE.

This, also, may be demonstrated by the student, by the aid of the previous theorems on intercepted arcs.

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PROBLEMS IN DRAWING.

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230. Problem.-Through a given point out of a circumference, to draw a tangent to the circumference.

Let A be the given point, and C the center of the given circle. Join AC. Bisect AC at the

point B (171). With B as a center and BC as a radius, describe a circumference. It will pass through C and A (153), and will cut the circumference in two points, D and E. Draw straight lines from A through D and E. AD and AE are both tangent to the given circumference.

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Join CD and CE. The angle CDA is inscribed in a semicircumference, and is therefore a right angle (225). Since AD is perpendicular to the radius CD, it is tangent to the circumference (178). AE is tangent for the same reasons.

231. Problem.—Upon a given chord to describe an arc which shall contain a given angle.

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Let AB be the chord, and C the angle. Make the angle DAB equal to C. At A erect a perpendicular to AD, and erect a perpendicular to AB at its center (172). Produce these till they meet at the point F (137). With F as a center, and FA as a radius, describe a circumference. Any angle inscribed in the arc BGHA will be equal to the given angle C.

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For AD, being perpendicular to the radius FA, is a tangent (178). Therefore, the angle BAD is measured by half of the arc AIB (226). But any angle contained in the arc AHGB is also measured by half of the same arc (222), and is therefore equal to BAD, which was made equal to C.

POSITIONS OF TWO CIRCUMFERENCES.

232. Theorem.-Two circumferences can not cut each other in more than two points.

For three points determine the position and extent of a circumference (149). Therefore, if two circumferences have three points common, they must coincide throughout.

233. Let us investigate the various positions which two circumferences may have with reference to each other.

Let A and B be the centers of two circles, and let these points be joined by a straight line, which therefore measures the distance between the centers.

First, suppose the sum of the radii to be less than AB.