E

105. Through a given point within a circle to draw a chord equal to a given chord. (§ 164.) What restriction is there on the position of the given point?

B

106. Through a given point to describe a circle of given radius tangent to a given straight line.

(Draw a || to the given line at a distance equal to the radius.)

107. To describe a circle of given radius tangent to two given circles.

(To describe a O of radius m tangent to two given whose radii are n and p, respectively.)

What restriction is there on the value of m?

Lo"

108. To describe a circle tangent to two given parallels, and passing through a given point.

What restriction is there on the position of the given point?

109. To describe a circle of given radius, tangent to a given line and a given circle.

(Draw a to the given line at a distance equal to the given radius.) 110. To construct a parallelogram, having given a side, an angle, and the diagonal drawn from the vertex of the angle.

111. In a given triangle to inscribe a rhombus, having one of its angles coincident with an angle of the triangle.

(Bisect the which is common to the ▲ and the rhombus.)

112. To describe a circle touching two given intersecting lines, one of them at a given point. (§ 169.)

113. In a given sector to inscribe a circle.

(The problem is the same as inscribing a O in ▲ O'CD.)

D

B

114. In a given right triangle to inscribe a square, having one of its angles coincident with the right angle of the triangle.

115. Through a vertex of a triangle to draw a straight line equally distant from the other vertices.

Х

116. Given the base, the altitude, and the vertical angle of a triangle, to construct the triangle. (§ 226.)

(Construct on the given base as a chord a segment which shall contain the given 2.)

117. Given the base of a triangle, its vertical angle, and the median drawn to the base, to construct the triangle.

118. To construct a triangle, having given the middle points of its sides.

120. Given the base, the altitude, and the radius of the circumscribed circle of a triangle, to construct the triangle.

(The centre of the circumscribed vertex equal to the radius of the O.)

lies at a distance from each

121. To draw common tangents to two given circles which do not intersect.

(To draw exterior common tangents, describe OAA' with its radius equal to the difference of the radii of the given .

To draw interior common tangents, describe OAA' with its radius equal to the sum of the radii of the given ©.)

Note. For additional exercises on Book II., see p. 224.

BOOK III.

THEORY OF PROPORTION. - SIMILAR

POLYGONS.

DEFINITIONS.

227. A Proportion is a statement that two ratios are equal.

228. The statement that the ratio of a to b is equal to the ratio of c to d, may be written in either of the forms

229. The first and fourth terms of a proportion are called the extremes, and the second and third terms the means.

The first and third terms are called the antecedents, and the second and fourth terms the consequents.

Thus, in the proportion a:c:d, a and d are the extremes, b and c the means, a and c the antecedents, and b and d the consequents.

230. If the means of a proportion are equal, either mean is called a mean proportional between the first and last terms, and the last term is called a third proportional to the first and second terms.

Thus, in the proportion a: b=b: c, b is a mean proportional between a and c, and c a third proportional to a and b.

231. A fourth proportional to three quantities is the fourth term of a proportion, whose first three terms are the three quantities taken in their order.

Thus, in the proportion a: b = c: d, d is a fourth proportional to a, b, and c.

PROP. I. THEOREM.

232. In any proportion, the product of the extremes is equal

233. Cor. The mean proportional between two quantities

is equal to the square root of their product.

Given the proportion a:bb: c.

To Prove

(1)

b = Vac.

234. (Converse of Prop. I.) If the product of two quantities is equal to the product of two others, one pair may be made the extremes, and the other pair the means, of a proportion.

Proof. Dividing both members of (1) by bd,

PROP. III. THEOREM.

235. In any proportion, the terms are in proportion by ALTERNATION; that is, the first term is to the third as the second term is to the fourth.

Given the proportion a: bc: d.

(1)

236. In any proportion, the terms are in proportion by INVERSION; that is, the second term is to the first as the fourth term is to the third.

Given the proportion a:bc: d.

(1)

237. In any proportion, the terms are in proportion by COMPOSITION; that is, the sum of the first two terms is to the first term as the sum of the last two terms is to the third term.

Given the proportion a: bc: d.

(1)

(?)