230. Exercise.-Two circles are tangent externally at A. The line of centres contains A, by § 224. Prove (1) that the perpendicular to the line of centres at A is a common tangent; (2) that it bisects the other two common tangents; and (3) that it is the locus of all points from which tangents drawn to the two circles are equal.

231. Exercise.-Find the locus of the middle points of all chords of a given length.

232. Exercise. If a straight line be drawn through the point of contact of two tangent circles forming chords, the radii drawn to the remaining extremities of these chords are parallel. Also, the tangents at these extremities are parallel. What two cases are possible?

PROBLEMS OF CONSTRUCTION

233. Exercise.-Draw a straight line tangent to a given circle and parallel to a given straight line.

234. Exercise.-Construct a right triangle, given the hypotenuse and an acute angle.

235. Exercise.-Construct a right triangle, given the hypotenuse and a side.

236. Exercise.--Construct a right triangle, given the hypotenuse and the distance of the hypotenuse from the vertex of the right angle.

237. Exercise.-Construct a circle tangent to a given. straight line and having its centre in a given point.

238. Exercise.-Construct a circumference having its centre in a given line and passing through two given points. 239. Exercise. Find the locus of the centres of all circles of given radius tangent to a given straight line.

240. Exercise.-Construct a circle of given radius tangent to two given straight lines.

241. Exercise.-Construct a circle of given radius tangent to two given circles.

242. Exercise.-Construct all the common tangents to two given circles.

Hint. For the external tangents draw a circle with radius equal to the difference of the radii of the given circies and its centre at the centre of the larger circle. Draw tangents to this circle from the centre of the smaller circle.

PLANE GEOMETRY

BOOK III

PROPORTION AND SIMILAR FIGURES

243. Def.-A proportion is an equality of ratios.

Thus, if the ratio is equal to the ratio

A

B

A C

equality

constitutes a proportion.

B D

C

D'

then the

This may also be written

A: BC: D, or A: B:: C: D,

and is read, A is to B as C is to D.

244. Def. The four magnitudes A, B, C, D are called the terms of the proportion.

245. Defs.-The first and last terms are the extremes, the second and third, the means.

246. Defs.-The first and third terms are called the antecedents, and the second and fourth, the consequents. 247. THEOREM. If four quantities are in proportion, their numerical measures are in proportion; and conversely.

where a, b, c, d are the numerical measures of

A, B, C, D, respectively.

[The ratio of two quantities is equal to the ratio of their numerical meas

248. Remark.--In order that the preceding theorems shall hold true, A and B must be quantities of the same kind, as two straight lines, or two angles, and C and D also of the same kind; but it is not necessary that A and B shall be of the same kind as C and D.

249. Def.-One variable quantity is said to be proportional to another, when any two values of the first have the same ratio as two corresponding values of the second.

arc.

Thus, Proposition XI., Book II., may be expressed :

An angle at the centre of a circle is proportional to its intercepted

By this we mean that the ratio of a given angle, as AOB, to some other angle, as A'O'B', is equal to the ratio of the corresponding arcs, AB and A'B'.

TRANSFORMATION OF PROPORTIONS

250. THEOREM. If four numbers are in proportion, the product of the extremes equals the product of the means.

Clear (1) of fractions, i. e., multiply both sides by bd, the product of the denominators of (1).

We have

ad=bc. (2)

Ax. 7 Q. E. D.

251. THEOREM. Conversely, if the product of two numbers equals the product of two others, either pair may be made the extremes and the other pair the means of a proportion.

Divide both sides of (2) by bd, the product of the denom

Dividing (2) by ac, the product of the denominators of (3), we obtain (3).

Q. E. D. Question. By dividing the equation ad = bc by the product of two of the letters, one being from each side, how many proportions in all can be obtained? Write them. If the equation be written be ad, how many can

=

be obtained, and how do they differ from the former set? 252. Remark.-The student has already noticed that the process by which equation (1) was obtained from (2) was the reverse of that by which (2) was obtained from (1). Also it is easy to see that (3) was obtained from (2) by a process the reverse of that by which (2) could have been obtained from (3). Now it is always much easier to see how an equation can be reduced to ad=bc than to see how it can be deduced from ad- bc. Since the latter is the reverse of the former, we have the following practical guide for obtaining a required equation from ad=bc: First see what processes would be necessary if you wished to reduce the equation to ad= bc; reverse these steps in order, and you have the method required.