92. Describe a circle with given radius to touch a given st. line and have its centre in another given st. line.

93. Describe a circle of given radius to pass through a given point and touch a given st. line.

94. Describe a circle to touch a given circle at a given point and a given st. line.

95. In a given st. line find a point such that the st. lines joining it to two given points may be (a) is, (b) make a given with each other.

96. Describe a circle of given radius to touch a given circle and a given st. line.

97. Describe a circle to touch a given circle and a given st. line at a given point.

98. Inscribe in a given circle a▲ one of whose sides shall be equal to a given st. line, and such that the other two may pass through two given points respectively.

99. Place a chord PQ in a circle so that it will pass through a given point O within the circle, and such that the difference between OP and OQ may be equal to a given st. line.

100. Find two points on the circumference of a given circle which shall be concyclic with two given points P and outside the circle.

101. Describe a square (EFGH) having given the point F and two points P and Q in the sides FE and EH respectively.

102. Describe a square (EFGH) having given the point G and two points P and Q in the sides FE and EH respectively.

103. Describe a square so that its sides shall pass respectively through four given points.

104. If three circles touch externally at P, Q, R and PQ and PR meet the circumference of QR at D and E, then DE is a diameter, and is || to the line joining the centres of the other two circles.

105. Two equal circles intersect so that the tangents at one of the points of intersection are 1s. Show that the square on the diameter is twice the square on the common chord.

106. LMN is a rt.-Ld A, L being the rt. 4, and LD is to MN. Show that LM is a tangent to the circle LDN.

107. PQ is a tangent to a circle and PRS a secant passing through the centre, QN is to PS. Show that QR bisects PQN.

108. LMN is a ▲ inscribed in a circle whose centre is O. Show that the radius OL makes the same L with LM that the L from L to MN makes with LN.

109. If two chords of a circle be 1, the sum of one pair of opposite intercepted arcs is equal to the sum of the other pair.

110. On the sides of a quadrilateral as diameters circles are described. Show that the common chords of every adjacent pair of circles is to the common chord of the remaining pair.

111. Two equal circles are so situated that the distance between their nearest points is less than the diameter of either circle. Show how to draw a st. line cutting them so as to be trisected by the circumferences.

112. LMN is a ▲ and D, E, F are the middle points of MN, NL and LM respectively; if LP is the perpendicular on MN, show that D, P, E, F are concyclic.

113. QR is a fixed chord of a circle and P a moveable point on the circumference. Find the locus of the intersection of the diagonals of the gm having PQ and QR for adjacent sides.

114. If a quadrilateral having two || sides is inscribed in a circle, show that the four perpendiculars from the middle point of an arc cut off by one of the || sides, to the two diagonals and to the nonparallel sides, are equal.

115. ABCD and A'B'C'D' are any rectangles inscribed in two concentric circles respectively. P is on the circumference of the former circle and P' on the latter. Prove PA+ PB + PC"2 + PD'2 = P'A2 + P'B2 + P'C2 + P'D2.

116. A point Y is taken in a radius of a circle whose centre is O; on OY as base an isosceles Δ ΧΟΥ is described having X on the circumference; XO and XY are produced to meet the circumference at D and Z respectively, and E is the point between D and Z where the perpendicular from O to OY cuts the circle. Show that the arc DE is one-third of arc EZ

RATIO AND PROPORTION

115. Definitions. The ratio of one magnitude to another of the same kind is the number of times that the first contains the second; or it is the part, or fraction, that the first magnitude is of the second.

Thus the ratio of one magnitude to another is the same as the measure of the first when the second is taken as the unit.

If a st. line is 5 cm. in length, the ratio of its length to the length of one centimetre is 5, that is, the st. line is to one centimetre as 5 is to 1.

If two st. lines A, B are respectively 8 inches and 3 inches in length, then the ratio of A to B is 8 to 3. The ratio of one magnitude A to another B is written either A

B

or A B.

A

B

When the form is used, the upper magnitude is called the numerator, and the lower the denominator; and when the form A B is used, the first magnitude is called the antecedent, and the second the consequent. The two magnitudes are called the terms of the ratio.

116. Definitions. Proportion is the equality of ratios, i.e., when two ratios are equal to each other, the four magnitudes are said to be in proportion.

The equality of the ratios of K to L and of M to N may be written in any one of the three forms:

The four magnitudes in a proportion are called proportionals.

The first and last are called the extremes, and the second and third are called the means.

The first two magnitudes of a proportion must be of the same kind, and the last two must be of the same kind; but the first two need not be of the same kind as the last two. Thus in the proportion D and E may be lengths of lines, while F and

D F

H

H are areas.

117. Definitions.-Three magnitudes are said to be in continued proportion, or in geometric progression, when the ratio of the first to the second equals the ratio of the second to the third.

Three magnitudes L, M, N, of the same kind, are in continued proportion, if

L M

M Ν

The second magnitude of a continued proportion is called the mean proportional, or geometric mean, of the other two.

118. Two magnitudes of the same kind are commensurable when each contains some common measure an integral number of times.

Two magnitudes of the same kind are incommensurable when there is no common measure, however small, contained in each of them an integral number of times.