CONSTRUCTIONS.

304. To divide a given line AB proportionally to another given line CD divided in C

E

FG D

lines parallel to BD', meeting AB in X, Y, Z, respectively. Then (274),

AX: XY: YZ : ZB = AE': E'F': F'G': G'D'=CE: EF: FG: GD.

If instead of a divided line we have numbers given, say 3, 7, 9, etc., we lay off on AD', AE'=3, E'F' 7, etc., and AB will be divided proportionally to the given numbers.

=

305. To find a fourth proportional to three given lines A, B, C.

A

B

C

A

H

E

Draw DE, DF, making any angle with each other. Upon DE lay off DG, GE, equal to A and B, respectively, and on DF lay off DH = C. Join GH, and through E draw EX parallel to GH, and meeting DF in X. Then

It is obvious that if B = C, we take GE =

(274),

DH, and we

obtain by this construction a third proportional to A and B.

X

306. To find a mean proportional between two lines, A, B. Upon an indefinite line lay off ED, CE, respectively equal to 4 and B. Upon CD as diameter describe a semicircle DXC; at E draw EX perpendicular to CD to meet the circumference in X. Then (312"),

E

D

307. DEFINITION. A straight line is said to be divided in extreme and mean ratio, when it is divided into two segments such that the greater segment is a mean proportional between the whole line and the lesser segment. Thus if AB is divided in C so that AB: AC AC: BC, then AB is divided in extreme and mean ratio.

308. To divide a line AB in extreme and mean ratio.

At B draw BC1 to AB, and

make BC = AB.

A

E

B

From Cas center, with radius CB, describe a circumf. Bde. Through A and C draw a secant meeting the circumference

in D and E.

From A as center, with radius AE, describe an arc EX meeting AB in X.

Then AB AX = AX: BX. For

... AB is a tangent (191), and AD a secant to OBDE, (Const.) AE: AB = AB : AD,

(303) (247)

.'. AX : BX = AB: AX, or AB : AX=AX: BX. (307)

309. To divide a given line AB harmonically in the ratio

equal to D. Join FB, GB, and draw EX, EY, parallel to GB and FB, respectively. Then,

AX: BX = AE: EG = C: D,

and AY: BY=AE: FE = C: D.

(275)

310. Upon a straight line A'B' to construct a polygon simi

AA'B'C' be similar to ▲ ABC (288). In like manner construct ▲ A'C'D' similar to ▲ ACD, and A'D'E' similar to ADE.

Polygon A'B'C'D'E' is similar to polygon ABCDE. (294)

311. Upon a given line AB, to describe a segment of a circle such that any angle inscribed

in it shall equal a given C.

At 4, make BAD=C (203); draw 40 perpendicular to AD at 4, and draw EO perpendicular to AB at its mid point E. From 0 the intersection of 40, EO, de

scribe arc ABX. ABX is the required segment.

X

Proof. BAD = C, is measured by arc ADB, as is

also any angle inscribed in BXA,

312. From a given point A, in or without a given circumference BCD, to draw a tangent to BCD.

Find o, the center of BCD, (173) and join 04. Then

1o. If A is in the circumference, draw XY perpendicular to OA at A. XY is tangent to BCD at A.

(191)

2o. If A is without the circumference, bisect OA in E. From E as center, with radius EA, describe a circumference AXY, cutting BCD in X and Y.

Join AX, AY; then AX, AY are tangents to BCD. (267, 191) Proof. Join ox, OY, and show that OX, OY are perpendicular to AX, AY, respectively.

EXERCISES.

THEOREMS.

396. The chords that join the near extremities of equal chords are parallel.

397. The opposite angles of a quadrilateral inscribed in a circle are supplementary.

DEFINITION. Three or more points are said to be concyclic if a circumference can be described through them. Thus the preceding theorem could be enunciated thus: A quadrilateral whose vertices are concyclic has its opposite angles supplementary.

398. If two opposite angles of a quadrilateral are supplementary, its vertices are concyclic.

399. If AB, CD, intersect in O, so that AO: OC = OD: OB, then A, B, C, D, are concyclic.

400. If OA, OD, are divided in B and C, respectively, so that OA: OD OC: OB, then A, B, C, D, are concyclic.

401. If an arc be divided into three equal parts by chords drawn from one extremity of the arc, the middle chord bisects the angle formed by the other two.

402. If, from any point of a circumference, a tangent and a chord be drawn, the perpendiculars upon these lines from the mid point of the intercepted arc are equal.

403. The diagonals of a trapezoid cut each other in the same ratio.

404. If through one of the points of intersection of two equal circles, any line be drawn to meet the circumferences, the extremities of this line are equally distant from the other point of intersection.

405. If, in a right triangle, the altitude upon the hypotenuse divides it in extreme and mean ratio, the lesser arm is equal to the farther segment.

406. In any right triangle, one arm is to the other as the difference of the hypotenuse and the second arm is to the intercept on the first arm between the right vertex and the bisector of the opposite acute angle.

407. The altitudes of a triangle are inversely proportional to the sides upon which they are drawn.

408. If from an angle of a parallelogram ABCD, a line be drawn cutting a diagonal in E and the sides in P and Q; respectively, then will AE be a mean proportional between PE and QE.

409. If from the extremities of a diameter, perpendiculars be drawn to any chord of the circle, the feet of these perpendiculars will be equally distant from the center.

410. Show that there may be two, but not more than two, similar triangles in the same segment of a circle.

411. If two circles are tangent externally, a common exterior tangent is a mean proportional between their diameters.

412. The chord drawn from the vertex of an inscribed equilateral triangle to any point in the opposite arc is equal to the sum of the chords drawn to that point from the other vertices.

413. If two circles are tangent externally, lines drawn through the point of contact to the circumferences are divided proportionally at the point of contact.

414. If a circle is tangent to another internally, all chords of the outer circle drawn from the point of contact are divided proportionally by the circumference of the inner circle.

Geom.-11