are unequally distant from the centre, they are unequal, and the chord at the less distance is the greater.

239. A straight line perpendicular to a radius at its extremity is a tangent to the circle.

240. Cor. 1. A tangent to a circle is perpendicular to the radius drawn to the point of contact.

241. Cor. 2. A perpendicular to a tangent at the point of contact passes through the centre of the circle.

242. Cor. 3. A perpendicular let fall from the centre of a circle upon a tangent to the circle passes through the point of contact.

243. Parallels intercept equal arcs on a circumference. 244. Through three points not in a straight line, one circumference, and only one, can be drawn.

245. Cor. Two circumferences can intersect in only two points.

246. The tangents to a circle drawn from an exterior point are equal, and make equal angles with the line joining the point to the centre.

249. If two circumferences intersect each other, the line of centres is perpendicular to their common chord at its middle point.

250. If two circumferences are tangent to each other, the line of centres passes through the point of contact.

260. If two variables are constantly equal and each approaches a limit, their limits are equal.

261. In the same circle, or equal circles, two angles at the centre have the same ratio as their intercepted arcs. 263. An inscribed angle is measured by one-half the arc intercepted between its sides.

264. Cor. 1. An angle inscribed in a semicircle is a right angle.

265. Cor. 2. An angle inscribed in a segment greater than a semicircle is an acute angle.

266. Cor. 3. An angle inscribed in a segment less than a semicircle is an obtuse angle.

267. Cor. 4. All angles inscribed in the same segment are equal.

268. An angle formed by two chords intersecting within the circumference is measured by one-half the sum of the intercepted arcs.

269. An angle formed by a tangent and a chord is measured by one-half the intercepted arc.

270. An angle formed by two secants, two tangents, or a tangent and a secant, intersecting without the circumference, is measured by one-half the difference of the intercepted arcs.

PROBLEMS OF CONSTRUCTION.

271. At a given point in a straight line, to erect a perpendicular to that line.

272. From a point without a straight line, to let fall a perpendicular upon that line.

273. To bisect a given straight line.

274. To bisect a given arc.

275. To bisect a given angle.

276. At a given point in a given straight line, to construct an angle equal to a given angle.

277. Two angles of a triangle being given, to find the third angle.

278. Through a given point, to draw a straight line parallel to a given straight line.

279. To divide a given straight line into equal parts. 280. Two sides and the included angle of a triangle being given, to construct the triangle.

281. A side and two angles of a triangle being given, to construct the triangle.

282. The three sides of a triangle being given, to construct the triangle.

283. Two sides of a triangle and the angle opposite one of them being given, to construct the triangle.

284. Two sides and an included angle of a parallelogram being given, to construct the parallelogram.

285. To circumscribe a circle about a given triangle. 287. Through a given point, to draw a tangent to a given circle.

288. To inscribe a circle in a given triangle.

290. Upon a given straight line, to describe a segment of a circle which shall contain a given angle.

291. To find the ratio of two commensurable straight lines.

BOOK III.

PROPORTION.

295. In every proportion the product of the extremes is equal to the product of the means.

296. A mean proportional between two quantities is equal to the square root of their product.

297. If the product of two quantities is equal to the product of two others, either two may be made the extremes of a proportion in which the other two are made the means.

298. If four quantities of the same kind are in proportion, they will be in proportion by alternation; that is, the first term will be to the third as the second to the fourth.

299. If four quantities are in proportion, they will be in proportion by inversion; that is, the second term will be to the first as the fourth to the third.

300. If four quantities are in proportion, they will be in proportion by composition; that is, the sum of the first two terms will be to the second term as the sum of the last two terms to the fourth term.

301. If four quantities are in proportion, they will be in proportion by division; that is, the difference of the first two terms will be to the second term as the difference of the last two terms to the fourth term.

302. In any proportion the terms are in proportion by composition and division; that is, the sum of the first two terms is to their difference as the sum of the last two terms to their difference.

303. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

304. The products of the corresponding terms of two or more proportions are in proportion.

305. Like powers, or like roots, of the terms of a proportion are in proportion.

307. Equimultiples of two quantities are in the same ratio as the quantities themselves.

THEOREMS.

309. If a line is drawn through two sides of a triangle parallel to the third side, it divides those sides proportionally.

310. Cor. 1. One side of a triangle is to either part cut off by a straight line parallel to the base as the other side is to the corresponding part.

311. Cor. 2. If two lines are cut by any number of parallels, the corresponding intercepts are proportional.

312. If a straight line divides two sides of a triangle proportionally, it is parallel to the third side.

313. The bisector of an angle of a triangle divides the opposite side into segments proportional to the other two sides.

314. The bisector of an exterior angle of a triangle meets the opposite side produced at a point the distances of which from the extremities of this side are proportional to the other two sides.

317. Cor. 1. The bisectors of an interior angle and an exterior angle at one vertex of a triangle divide the opposite side harmonically.

318. Cor. 2. If the points M and M' divide the line AB harmonically, the points A and B divide the line MM' harmonically.

321. Two mutually equiangular triangles are similar. 322. Cor. 1. Two triangles are similar if two angles of the one are equal respectively to two angles of the other.