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59. Through a given point P between two given straight lines AB, AC, draw a straight line which shall be termi nated by AB, AC, and bisected in P.

60. If P be outside the lines AB, AC, draw PDE meeting AB, AC in D, E so that PD equals DE.

61. Find points D, E in the sides AB, AC respectively of the triangle ABC, so that DE may be parallel to BC and equal to BD.

62. Draw a straight line DE parallel to the base BC of the triangle ABC and meeting AB, AC in D and E, so that DE equals the sum of BD, CE.

63. From a given point without a given straight line draw a straight line making a given angle with this given line.

64. AB, AC are two straight lines, AE is an intermediate straight line. Show how to draw the straight lines which are terminated by AB, AC, and bisected by AE.

Take any pt. E in AE, draw EF to AC meeting AB in F; from FB cut off FG = AF, etc.

65. Construct a triangle, having given a median and the two angles into which the angle is divided by that median.

66. Construct a parallelogram, having given (1) two diagonals and the angle between them, and (2) one side, one diagonal, and the angle between the diagonals.

67. From a given point O draw three straight lines OA, OB, OC of given lengths so that A, B, C may be collinear, and AB equal to BC.

Construct a ▲ OAD having the sides OA, AD the two outside lengths, and OD double the middle, etc.

68. In a given straight line find a point whose distance from a given point in the line may be equal to its distance from another given straight line.

Let P be the given pt. in the given line XY, AB the other given line. Draw PC to AB, bisect / XPC, etc.

69. Construct a triangle, having given the base, the vertical angle, and (1) the sum, or (2) the difference of the sides,

Construct a right triangle having given:

70. The hypotenuse and the sum of the sides.

71. The hypotenuse and the difference of the sides. 72. The hypotenuse and the perpendicular from the right angle on it.

73. The perimeter and an angle.

74. Construct an isosceles triangle, having the vertical angle equal to four times each of the base angles.

75. Construct an isosceles triangle, having one-third of each angle at the base equal to half the vertical angle.

CIRCLES.

76. Through a given point inside a circle which is not the centre, draw a chord which is bisected at that point.

77. With a given point as centre describe a circle which shall intersect a given circle at the ends of a diameter.

78. Describe a circle which shall pass through a given point outside or inside a given circle and touch it at a given point.

79. Describe a circle with a given centre which shall touch a given circle. How many such circles can be drawn?

80. Through a given point inside a given circle draw two equal chords which shall contain an angle equal to a given angle.

81. Describe a circle which shall touch a given circle at a given point, and also touch a given straight line.

Let C be the cent. of O, P the point. Draw the diam. ABCD to given line XY, meeting the in A; join PA, PB, and produce them to meet XY in E, F, etc.

82. Describe a circle passing through a given point and touching a given straight line at a given point.

83. With a given point outside a given circle as centre, describe a circle which shall cut the given circle orthogonally (228).

84. Given two points A, B on a circle, and a fixed straight line through A. Draw through B a straight line.

cutting the circle in C, and the fixed line in D, so that AC shall be equal to CD.

Let the fixed line meet the O in F; bisect the arc AB in E, draw EC to AF, etc.

85. Describe a circle which shall touch a given straight line at a given point, and pass through another given point not in the line.

86. Construct a triangle, having given the base, the vertical angle, and the median drawn from the vertical angle.

87. ABC is a given straight line: find a point P such that each of the angles APB, BPC may be equal to a given angle.

88. Find the point inside a given triangle at which the sides subtend equal angles.

89. About a given circle to describe a triangle equiangular to a given triangle.

On two sides describe segments of Os containing an = 3 of 2 rt. Zs, etc.

90. If the escribed circles of the triangle ABC (272) touch BC, CA, AB externally in D, E, F respectively, prove that BD = EA, CD = AF, CE = BF.

BOOK III.

RATIO AND PROPORTION. SIMILAR

FIGURES.

DEFINITIONS.

277. Four quantities are said to be in proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth.

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then, A, B, C, D are said to be in proportion. The proportion is written

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NOTE. The first form is preferable, and the one most generally used in the higher mathematics; but the second form is the more usual one in elementary works.

Let a and b denote the numerical measures of A and B (229), and c and d the numerical measures of C and D. Then, since the ratio of two quantities is the same as the ratio of their numerical measures (230),

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Hence, if four quantities A, B, C, D, are in proportion, their numerical measures a, b, c, d, are in proportion; that is,

a b c d

Conversely, if the numerical measures of four quantities A,B,C,D, are in proportion, the quantities themselves are in proportion; that is,

A: B CD,

when A and B are quantities of one kind, and C and D are quantities of one kind, though the latter kind may be different from the former.

That is, all four quantities may be of the same kind, as, for instance, four straight lines, four surfaces, four angles, and so on; but the quantities in each pair must be of the same kind.

The magnitudes we meet with in Geometry are more often incommensurable (232) than commensurable.

The preceding reasoning does not apply directly to the case in which two quantities are incommensurable, but it be extended to this case.

may

278. To find the greatest common measure of two quantities.

Let there be two quantities, as,

for instance, the two straight lines AB, CD.

Apply the smaller CD to the

E GB

C F D

greater AB, as many times as possible, suppose twice, with a remainder EB.

Apply the remainder EB to CD as many times as possible, suppose once, with a remainder FD.

Apply the second remainder FD to EB as many times as possible, suppose once, with a remainder GB.

Apply the third remainder GB to FD as many times as possible.

This process will terminate only if a remainder is found which is contained an exact number of times in the preceding one; and if it so terminates the two given lines are commensurable, and the last remainder will be their greatest common measure,

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