83. State and prove the converse of Prop. XLV.

84. State and prove the converse of Ex. 65, p. 66.

85. Prove Prop. XXVI. by drawing through B a parallel to AC. 86. Prove Prop. XXX. by drawing CD to the middle point of AB. 87. Prove Prop. XXXI. by drawing CD so as to bisect / ACB. 88. The middle point of the hypotenuse of a right triangle is equally distant from the vertices of the triangle.

89. The bisectors of the angles of a rectangle form a square.

90. If D is the middle point of the side BC of the triangle ABC, and BE and CF are the perpendiculars from B and C to AD, produced if necessary, prove that BE = CF.

91. The angle at the vertex of an isosceles triangle ABC is equal to twice the sum of the equal angles B and C. If CD be drawn perpendicular to BC, meeting BA produced at D, prove that the triangle ACD is equilateral.

=

92. The bisector of the vertical angle A of an equilateral triangle ABC is produced to D, so that AD AB. If BD and CD be drawn, prove that ▲ BDC is 30° or 150°, according as D lies above or below the base.

93. If the angle B of the triangle ABC is greater than the angle C, and BD be drawn to AC making AD = AB, prove that

2 ADB = } (B + C ), and ≤ CBD = } (B — C').

94. The sum of the lines drawn from any point within a triangle to the vertices is greater than the half-sum of the three sides. (§ 61.)

95. The sum of the lines drawn from any point within a triangle to the vertices is less than the sum of the three sides.

96. How many sides are there in the polygon the sum of whose interior angles exceeds the sum of its exterior angles by 540° ?

97. If D, E, and F are points on the sides AB, BC, and CA of an equilateral triangle ABC, such that AD BE CF, prove that the figure DEF is an equilateral triangle.

98. If E, F, G, and H are points on the sides AB, BC, CD, and DA of a parallelogram ABCD, such that AE = CG and BF = DHI, prove that the figure EFGH is a parallelogram.

99. If E, F, G, and H are points on the sides AB, BC, CD, and DA of a square ABCD, such that AE= BF = CG = DH, prove that the figure EFGH is a square.

100. If on the diagonal BD of a square ABCD a distance BE is taken equal to AB, and EF is drawn perpendicular to BD, meeting AD at F, prove that AF EF = ED.

101. Prove the theorem of § 127 by drawing lines from any point within the polygon to the vertices.

102. State and prove the converse of Ex. 68, p. 66.

103. State and prove the converse of Prop. XXXVIII. 104. State and prove the converse of Ex. 76, p. 67.

105. If AD is the perpendicular from the vertex of the right angle to the hypotenuse of the right triangle ABC, and AE is the bisector of the angle A, prove that DAE is equal to one-half the difference of the angles B and C.

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

107. 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 that the triangle OAB is isosceles.

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

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

110. If AD is the perpendicular from the vertex of the right angle to the hypotenuse of the right triangle ABC, and E is the middle point of BC, prove that DAE is equal to the difference of the

angles B and C.

111. If one acute angle of a right triangle is double the other, the 'hypotenuse is double the shortest leg.

112. If AD be drawn from the vertex of the right angle to the hypotenuse of the right triangle ABC so as to make ≤ DAC = ≤ C, it bisects the hypotenuse.

113. If D is the middle point of the side BC of the triangle ABC, prove that AD>(AB+ AC — BC). (§ 62.)

NOTE. For additional exercises on Book I., see p. 221.

BOOK II.

THE CIRCLE.

DEFINITIONS.

B

142. A circle 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. Any portion of the circumference, as

AB, is called an arc.

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

as OA.

D

C

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 Ꭶ

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; and 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 the chord AB subtends the arcs AMB and and ACDB.

When the arc subtended by a chord is spoken of, that are which is less than a

M

A

B

N

semi-circumference is understood, unless the contrary is specified.

A segment of a circle is the portion included between an are 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 extremities of the subtending chord.

Thus, the angle B is inscribed in the segment ABC.

B

149. A straight line is said to touch, or be tangent to, 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

Ө

A

intersects the circumference in two points; as CD.

-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 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 its sides are tangent to the circle; as EFGH.

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

PROPOSITION I. THEOREM.

152. Every diameter bisects the circle and its circumference.

Let AC be a diameter of the circle ABCD.

To prove that AC bisects the circle and its circumference.