PROBLEMS OF DEMONSTRATION

223. Defs. Two circles are tangent which touch at but one point. They may be tangent internally, so that one circle is within the other; or externally, so that each is without the other.

224. Exercise.—The straight line joining the centres of two circles tangent externally passes through the point of tangency.

X

Hint. Suppose 00′ not through T, and prove 00' greater than and also less than the sum of the radii.

225. Exercise. The straight line joining the centres of two circles internally tangent passes through the point of tangency.

Hint.-If not, prove the distance between centres greater than and also

less than the difference of the radii.

226. Defs.-If each of two circles is entirely without the other, four common tangents can be drawn. Two of these are called external, and two internal. An external tangent is one such that the two circles lie on the same side of it; an internal tangent is one such that the two circles lie on opposite sides of it.

Question.-In case the two circles are themselves tangent externally, how many common tangents of each kind can be drawn? In case the two circles overlap? In case they are tangent internally? In case one is within the other?

227. Exercise.-The two common external tangents to two circles meet the line joining their centres in the same point. Also the two common internal tangents meet the line of centres in the same point.

228. Exercise.—The sum of two opposite sides of a quadrilateral circumscribed about a circle is equal to the sum of the other two sides (§ 176).

229. Exercise.-The sum of two opposite angles of a quadrilateral inscribed in a circle is equal to the sum of the other two angles, and is equal to two right angles.

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

C

then the

D

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.