Ex. 83. The line jaining the middle points of the diagonals of a trapezoid is equal to half the difference of the bases.

A BFG: =▲ DFC. (Why ?) .. EF = AG (§ 189).

Ex. 84. In an isosceles trapezoid each base makes equal angles with the legs.

Draw CE to DB. CE = DB. (Why?) ZA=LCEA, 2B=LCEA. C and D have equal supplements.

Ex. 85. If the angles at the base of a trapezoid are equal, the other angles are equal, and the trapezoid is isosceles.

Ex. 86. In an isosceles trapezoid the opposite angles are supplementary. C = 2D (Ex. 84).

Ex. 87. The diagonals of an isosceles trapezoid are equal.

Prove ▲ ACDA BDC.

Ex. 88. If the diagonals of a trapezoid are equal, the trapezoid is isosceles.

Ex. 89. If from the diagonal DB, of a square ABCD, BE is cut off equal to BC, and EF is drawn perpendicular to BD meeting DC at F, then DE is equal to EF and also to FC.

LEDF = 45°, and ▲ DFE = 45°; and DE = EF. Rt. ▲ BEF = rt. ▲ BCF (§ 151); and EF = FC.

D

E

Ex. 90. Two angles whose sides are perpendicular, each to each, are either equal or supplementary.

BOOK II.

THE CIRCLE.

DEFINITIONS.

216. A circle is a portion of a plane bounded by a curved line, all points of which are equally distant from a point within I called the centre. The bounding line is called the circumference

of the circle.

217. A radius is a straight line from the centre to the circumference; and a diameter is a straight line through the centre, with its ends in the circumference.

By the definition of a circle, all its radii are equal. All its diameters are equal, since a diameter is equal to two radii.

218. Postulate. A circumference can be described from any point as a centre, with any given radius.

219. A secant is a straight line of unlimited length which intersects the circumference in two points; as, AD (Fig. 1). 220. A tangent is a straight line of unlimited length which has one point, and only one, in common with the circumference; as, BC (Fig. 1). In this case the circle is said to be tangent to the straight line. The common point is called the point of contact, or point of tangency.

B

FIG. 1.

221. Two circles are tangent to each other, if both are tangent to a straight line at the same point; and are said to be tangent internally or externally, according as one circle lies wholly within or without the other.

222. An arc is any part of the circumference; as, BC (Fig. 3). Half a circumference is called a semicircumference. Two arcs are called conjugate arcs, if their sum is a circumference.

223. A chord is a straight line that has its extremities in the circumference; as, the straight line BC (Fig. 3).

224. A chord subtends two conjugate arcs. If the arcs are unequal, the less is called the minor arc, and the greater the major arc. A minor arc is generally called simply an arc.

225. A segment of a circle is a portion of the circle bounded by an arc and its chord (Fig. 2).

226. A semicircle is a segment equal to half the circle (Fig. 2).

227. A sector of a circle is a portion of the circle bounded by two radii and the arc which they intercept. The angle included by the radii is called the angle of the sector (Fig. 2).

228. A quadrant is a sector equal to a quarter of the circle (Fig. 2).

229. An angle is called a central angle, if its vertex is at the centre and its sides are radii of the circle; as, ZAOD (Fig. 2).

230. An angle is called an inscribed angle, if its vertex is in the circumference and its sides are chords; as, ABC (Fig. 3). An angle is inscribed in a segment, if its vertex is in the arc of the segment and its sides pass through the extremities of the arc.

231. A polygon is inscribed in a circle, if its sides are chords; and a circle is circumscribed about a polygon, if all the vertices of the polygon are in the circumference (Fig. 3).

232. A circle is inscribed in a polygon, if the sides of the polygon are tangent to the circle; and a polygon is circumscribed about a circle, if its sides are tangents (Fig. 4).

233. Two circles are equal, if they have equal radii. For they will coincide, if their centres are made to coincide. CONVERSELY: Two equal circles have equal radii.

234. Two circles are concentric, if they have the same centre.

ARCS, CHORDS, AND TANGENTS.

PROPOSITION I. THEOREM.

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

Let HK be any line meeting the circumference HKM in H and K.

To prove that HK cannot meet the circumference in any other point.

Proof. If possible, let HK meet the circumference in P.

§ 217

Then the radii OH, OP, and OK are equal.

.. P does not lie in the straight line HK. .. HK meets the circumference in only two points.

§ 102

Q. E.D.

PROPOSITION II. THEOREM.

236. In the same circle or in equal circles, equal central angles intercept equal arcs; and of two unequal central angles the greater intercepts the greater arc.

C

B

P

In the equal circles whose centres are O and O', let the angles AOB and A'O'B' be equal, and angle AOC be greater than angle A′O ́B ́. To prove that

Proof. 1. Place the OA'B'P' on the ZA'O'B' shall coincide with its equal, the

1. arc AB = arc A'B';

2. arc AC arc A'B'.

ABP so that the
AOB.

§ 233

§ 216

Then A' falls on A, and B' on B.

... arc A'B' coincides with arc AB.

2. Since the AOC is greater than the A'O'B',

it is greater than the AOB, the equal of the A'O'B'.

Therefore, OC falls without the AOB.

.. arc AC > arc AB.

arc AC > arc A'B', the equal of arc AB.

Ax. 8

Q. E. D.

237. CONVERSELY: In the same circle or in equal circles, equal ares subtend equal central angles; and of two unequal arcs the greater subtends the greater central angle.