2. Since X is equidistant from C and D, it should be found in the locus of all points that are equidistant from c and D. Now that locus is (213) the perpendicular at the mid point of the line joining C and D ;

.. X is found at the intersection of that locus with AB.

SYNTHESIS. Join CD. Draw the FH at the mid point of CD, and let FH intersect AB in X. From X as center, with a radius equal to the distance XC, describe the circumference CDE. CDE is the required circumference.

Since X is a point in the at the mid point of CD,
X is equidistant from C and D,

(213)

.. a circumference passing through C will also pass through D,

(164)

.. a circumference having its center in AB has been described through C and D.

Q.E.F.

SCHOLIUM. The problem becomes impossible in a certain In what case?

case.

EXERCISES.

PROBLEMS.

240. Construct an isosceles triangle having its sides each double the length of the base.

241. Upon a given base AB, construct a right isosceles triangle.

242. With a given line AB as diagonal, construct a square.

243. Construct an equilateral triangle having a given altitude AB.

254. Three lines being given diverging from a point, draw a fourth line cutting them so that the intercepted segments shall be equal.

255. Construct an isosceles right triangle, the sum of the hypotenuse and a side being given.

256. Construct an isosceles right triangle, the difference of the hypotenuse and a side being given.

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

258. Construct an isosceles triangle of given altitude, whose sides pass through two given points, and whose vertex is in a given straight line.

Construct an isosceles triangle, having given:

259. The base and the vertical angle.

260. The base and a base angle.

261. An arm and the vertical angle.

Construct a triangle, having given :

262. Two sides and the included angle.

263. The base and the base angles.

=

Ꮓ

264. The three sides, AB, AC, BC, such that AC AB, and BC= AC.

265. Construct an isosceles right triangle, having given the sum and the difference of the hypotenuse and an arm.

HINT. It is useful to remember that A and B, being any two magnitudes,

(A + B) + ( A − B) = 2 A ; (A + B) − (A − B) = 2 B.

266. Construct a right triangle, having given an arm and the altitude from the right angle upon the hypotenuse.

267. Construct a right triangle, having given the hypotenuse and the difference of the other sides.

Construct a parallelogram, having given:

268. Two adjacent sides and a diagonal. 269. A side and both diagonals.

270. Both diagonals and their included angle.

Construct a trapezoid, having given :

271. The four sides.

272. The parallel sides and the diagonals.

273. The parallel sides, a diagonal, and the angle formed by the diagonals.

274. Through a point within a circle, draw a chord that is bisected in that point, and show it is the least chord through that point.

275. The position and magnitude of two chords of a circle being given, describe the circle.

276. In a given circle, draw a chord whose length is double its distance from the center.

277. Draw that diameter of a given circle, which, being produced, meets a given line at a given distance from the center. When is this impossible?

278. Describe a circle with given radius, to touch a given line in a given point. How many such circles can be described?

279. Describe a circle of given radius to touch two intersecting lines. How many such circles can be described ?

280. Describe a circle touching two intersecting lines at a given distance from their intersection. How many such circles can be described?

281. Describe a circumference passing through a given point, and touching a given line in a given point.

282. Describe a circumference touching two given lines, and passing through two given points between those lines.

283. From a given center, describe a circumference that bisects a given circumference.

284. With a given radius, describe a circle touching two given circles.

BOOK III.

RATIO. PROPORTION.

LIMITS.

MEASUREMENT.

For many purposes, as in the propositions thus far considered, it is sufficient to prove, in regard to two given magnitudes, that they are equal or unequal. Thus we proved, in Prop. XXIV. (99), that PA = PB, and PC > PA. We have now to consider how to proceed when we wish to estimate exactly the relative greatness of given magnitudes.

220. To measure a magnitude is to find out how many times it contains another magnitude of the same kind.

Thus we measure a line by finding how many times i contains another line called the unit of length, or linear unit. This unit may be either a standard unit, as an inch, a meter, etc., or a unit found by dividing a line into any desired number of equal parts, as in Prop. XIX. (207).

221. A quantity is a magnitude conceived as consisting of some number of equal parts.

Thus AB, regarded merely as a line, is a magnitude; but when thought of, or referred to, as 22 millimeters, or x linear units of any kind, it is a quantity meas- A

B

ured by millimeters, or some other unit. The angle BAC, again, if referred to as an angle of 31° 15′ 47.2", is a quantity measured by tenths of seconds.