98. State and prove the converse of Ex. 75, p. 68. (Fig. of Ex. 78. Prove & ACF and BDE equal.)

99. D is any point in base BC of isosceles triangle ABC. The side AC is produced from C to E, so that CE = CD, and DE is drawn meeting AB at F. Prove Z AFE = 3 ZAEF.

(ZAFE is an ext. Z of ▲ BFD.)

100. If ABC and ABD are two triangles on the same base and on the same side of it, such that AC = BD and AD BC, and AD and BC intersect at O, prove triangle OAB isosceles.

=

101. If D is the middle point of side AC of equilateral triangle ABC, and DE be drawn perpendicular to BC, prove EC: BC.

=

(Draw DF to the middle point of BC.)

102. If in parallelogram ABCD, E and F are the middle points of sides BC and AD, respectively, prove that lines AE and CF trisect diagonal BD.

F

B

D

A

(By § 131, AE bisects BH, and CF bisects DG.)

103. If CD is the perpendicular from C to the hypotenuse of right triangle ABC, and E is the middle point of AB, prove DCE equal to the difference of angles A and B. (Ex. 83.)

B

ய

D

104. If one acute angle of a right triangle is double the other, the hypotenuse is double the shorter leg.

(Fig. of Ex. 86. Draw CA to middle point of BD.)

105. If AC be drawn from the vertex of the right angle to the hypotenuse of right triangle BCD so as to make

ACD = 2 D, it

bisects the hypotenuse.

(Fig. of Ex. 74. Prove ABC isosceles.)

106. If D is the middle point of side BC of triangle ABC, prove AD> } (AB + AC − BC). (§ 62.)

Note. For additional exercises on Book I., see p. 220.

Воок II.

THE CIRCLE.

DEFINITIONS.

B

142. A circle (O) is a portion of a plane bounded by a curve called a circumference, all points of which are equally distant from a point within, called the centre; as ABCD.

An are is any portion of the circumference; as AB.

A radius is a straight line drawn from the centre to the circumference; as OA.

A

с

D

A diameter is a straight line drawn through the centre, having its extremities in the circumference; as AC.

143. It follows from the definition of § 142 that

All radii of a circle are equal.

Also, all its diameters are equal, since each is the sum of two radii.

144. Two circles are equal when their radii are equal. For they can evidently be applied one to the other so that their circumferences shall coincide throughout.

145. Conversely, the radii of equal circles are equal.

146. A semi-circumference is an arc equal to one-half the circumference.

A quadrant is an arc equal to one-fourth the circumference. Concentric circles are circles having the same centre.

147. A chord is a straight line joining the extremities of an arc; as AB.

The arc is said to be subtended by its chord.

Every chord subtends two arcs; thus chord AB subtends arcs AMB and ACDB.

M

A

B

N

D

When the arc subtended by a chord is spoken of, that are which is less than a semi-circumference is understood, unless the contrary is specified.

A segment of a circle is the portion included between an arc and its chord; as AMBN.

A semicircle is a segment equal to one-half the circle.

A sector of a circle is the portion included between an arc and the radii drawn to its extremities; as OCD.

148. A central angle is an angle whose vertex is at the centre, and whose sides are radii; as AOC.

An inscribed angle is an angle whose vertex is on the circumference, and whose sides are chords; as ABC.

An angle is said to be inscribed in a segment when its vertex is on the arc of the segment, and its sides pass through the A extremities of the subtending chord.

Thus, angle B is inscribed in segment ABC.

B

149. A straight line is said to be tangent to, or touch, a circle when it has but one point in common with the circumference; as AB.

In such a 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. A secant is a straight line which intersects the circumference in two points; as CD.

A

-B

150. Two circles are said to be tangent to each other when they are both tangent to the same straight line at the same point.

They are said to be tangent internally or externally according as one circle lies entirely within or entirely without the other.

A common tangent to two circles is a straight line which is tangent to both of them.

151. A polygon is said to be inscribed in a circle when all its vertices lie on the circumference; as ABCD.

In such a case, the circle is said to be circumscribed about the polygon.

A polygon is said to be inscriptible when it can be inscribed in a circle.

A polygon is said to be circumscribed about a circle when all its sides are tangent to the circle; as EFGH.

In such a case, the circle is said to be inscribed in the polygon.

PROP. I. THEOREM.

152. Every diameter bisects the circle and its circumference.

To Prove that AC bisects the O, and its circumference.

Proof. Superpose segment ABC upon segment ADC, by folding it over about AC as an axis.

Then, arc ABC will coincide with arc ADC; for otherwise there would be points of the circumference unequally distant from the centre.

Hence, segments ABC and ADC coincide throughout, and are equal.

Therefore, AC bisects the O, and its circumference.

PROP. II. THEOREM.

153. A straight line cannot intersect a circumference at more than two points.

Given O the centre of a O, and MN any str. line.

To Prove that MN cannot intersect the circumference at more than two points.

ОА = OB=OC.

Proof. If possible, let MN intersect the circumference at three points, A, B, and C; draw radii OA, OB, and OC. Then, We should then have three equal str. lines drawn from a point to a str. line.

(§ 143)

But this is impossible; for it follows from § 49 that not more than two equal str. lines can be drawn from a point to a str. line.

Hence, MN cannot intersect the circumference at more than two points.

Ex. 1. What is the locus of points at a given distance from a given point ?