330. COR. Conversely, if AD meets BC produced so that

331. Defs.-The line AB is divided internally at C, when this point is between the extremities of the line; CA and CB are the segments into which it is divided.

AB is divided externally at C', when this point is on the line produced. The segments are C'A and C'B.

In each case the segments are the distances from the point of division to the extremities of the line. The line is the sum of the internal segments, and the difference of the external segments.

332. A line is divided harmonically, when it is divided internally and externally in the same ratio.

333. Exercise.-Prove that the bisectors of the interior and exterior angles at one of the vertices of a triangle divide the opposite side harmonically (see figure below).

334. Exercise.-If AD and AE bisect the angles at A, prove also that ED is divided harmonically at C and B.

E

C D

Hint.-Alternate the proportion found in § 333.

B

335. Def.-A straight line is divided in extreme and mean ratio when one of its segments is a mean proportional between the whole line and the other segment.

336. CONSTRUCTION. To divide a given straight line in extreme and mean ratio.

REQUIRED to divide it in extreme and mean ratio.

At B draw the perpendicular BO equal to one half AB. With the centre O and radius OB describe a circumference, and draw AO, cutting the circumference in D and D'. On AB lay off AC=AD, and extend BA to C', making AC' AD'.

Then AB is divided in extreme and mean ratio, internally at C, and externally at C'.

AD'-AB-AD' — DD' = AD=AC, and AB—AD=BC.

Hence AB is divided internally at C in extreme and mean

Hence AB is divided externally at C' in extreme and mean ratio.

Q. E. F.

337. Remark.-AC and AC' may be computed in terms of AB as follows:

Likewise AC' = AD' = AO+OD'=AO+

But

4

4

AO=AB2 + (42)2 = AB′+ AB'.

Whence, extracting the square root,

PROBLEMS OF DEMONSTRATION

338. Exercise.-The point of intersection of the internal tangents to two circles divides the line of centres internally into parts whose ratio equals the ratio of the radii.

339. Exercise.—The point of intersection of the external tangents to two circles divides the line of centres externally into parts whose ratio equals the ratio of the radii.

340. Exercise.-The points of intersection of the internal and external tangents to two circles divide the line of centres harmonically.

341. Exercise.—If through the centres of two circles two parallel radii are drawn in the same direction, the straight line joining their extremities will pass through the intersection of the external tangents.

342. Exercise.—If through the centres of two circles two parallel radii are drawn in opposite directions, the straight line joining their extremities will pass through the intersection of the internal tangents.

343. Exercise.-If through the intersection of the external or of the internal tangents to two circles a secant is drawn, the radii to the points of intersection will be parallel in pairs.

344. Exercise.-Give methods for drawing the common tangents to two circles depending on §§ 341, 342.

345. Exercise.—A triangle ABC is inscribed in a circle to which a second circle is externally tangent at A. If AB and AC are produced till they meet the second circumference at M and N, the triangles ABC and AMN are similar. S$ 205, 275

346. Exercise-The perpendiculars from any two vertices of a triangle on the opposite sides are inversely proportional to those sides. 8276 347. Exercise. If two circles are tangent internally, all chords of the greater drawn from the point of contact are divided proportionally by the circumference of the smaller. Hint.-Apply §§ 202, 225, 276.

348. Exercise.-If from P, a point in a circumference, any chords, PA, PB, PC, are drawn, and these chords are cut in a, b, c, respectively, by any straight line parallel to the tangent at P, then PA × Pa=PB × Pb = PC × Pc.

Hint.-Let one chord pass through centre. Join its extremity to any other chord and apply §§ 202, 276.

349. Exercise.-On a common base AB are two triangles, ABC and ABC', whose vertices C and C' lie in a straight line parallel to AB. If a second parallel to AB cuts AC and BC in M and N, and AC' and BC' in M' and N', then MN= M'N'. $275

350. Exercise.-If at the extremities of BC, the hypotenuse of a right triangle ABC, perpendiculars to the hypotenuse are drawn intersecting AB produced in M and AC produced in N, then AB AM AN AC

351. Exercise. The difference of the squares of two sides of any triangle is equal to the difference of the squares of the projections of these sides on the third side.

$317