30.* Two points A and B are distant a from one another. A point P k. BP. Prove that the diameter of the circle which

moves so that AP =

31.* If ABCD is a cyclic quadrilateral, and AB = a, AC = c, DB = b, DC=d, prove that

*

(ab - cd) (bc

ac bd

ad)

ad)

,

32. Given any four straight lines, show how to form them into a cyclic quadrilateral. [Three solutions.]

33.* If D be the diameter of the circle circumscribing a quadrilateral whose sides are a, b, c, d, and ▲ be its area, prove that

CHAPTER XXIII.

PROBLEMS.

324. PROBLEM. To divide a straight line proportionally to a given divided line.

Required to divide the line AB proportionally to the divided line CDEF (Fig. 460).

... the parallels AZ, MH, NK, BL meet the transversals AB, AL,

COROLLARY 1.-To divide a given line AN internally in the ratio : m.

Draw AX at any convenient angle, and (choosing any convenient units) mark off from it AH = 1 units and HK = m units (Fig. 460). Draw HM | KN to meet AN at M.

Then AM: MN AH: HK = 1: m.

=

COROLLARY 2.-To divide a given line AB externally in the ratio : m.

Draw AX at any convenient angle, and (choosing any convenient units) mark off from it AH = 1 units and HK = m units, drawing AH and HK in opposite senses. (In Fig. 4617 > m, and in Fig. 462 l < m). Join KB. Draw HC || KB to meet AB at C.

Required to find a fourth prop. to L, M, and W (Fig. 463).

L

M.

N

FIG. 463.

A

Cons. Draw two straight lines OA and OB making any convenient angle AOB.

From OA cut off OC

from OB cut off OE

=

=

L, and CD

M;

N.

Through D draw DF || CE meeting OB in F.

Then EF is the fourth proportional to L, M, and N.

COROLLARY 1.—To find the third proportional to two lines. To find the third proportional to L and M (Fig. 463):Make OCL, CDM, OEM, and draw DF || CE.

Then

L:M

=

M: EF.

:

COROLLARY 2.-To find a line which shall be in a given ratio to a given line.

To find a line which shall be to L in the ratio h: k (Fig. 463):

Make OEL; also using any convenient units make OC = k units, CD = h units. Draw DF || CE.

Then EFL

=

h:k.

326. PROBLEM. To find the mean proportional to two lines. Required to find the mean proportional between ▲ and M (Fig. 464).

L

M

B

FIG. 464.

Take any straight line, and in it cut off, L and DC

Cons. BD

M.

On BC as diameter describe the semicircle BAC.

From D draw DAL BC, meeting Oce in A.

Then DA is the mean proportional between ▲ and M.

Proof. Join BA and AC.

... BAC is a semicircle, .. BAC = R,

Also AD is drawn from the right angle 1 BC;

Hence BDA | ADC, and BD: DA = DA: DC,

that is

L: DA

NOTE. -Since DC is the third proportional to BD, AD, and BC is the third proportional to BD, BA, this figure provides also two other distinct methods for finding the third proportional to two given lines. The student should work out both of these suggestions.

EXERCISES CXXII.

PRACTICAL GEOMETRY.

1. Draw a straight line 2" long, and divide it in the ratio of the side of a square to its diagonal, Measure the longer segment.

2. Draw a straight line 2 cms. long, and divide it externally in the ratio of the side of an equilateral triangle to its altitude. Measure the longer segment.

Construct and measure

cms.

Bisect BC at D.

3. Draw an equilateral triangle of side 1.2". the third proportional to its altitude and base. 4. Draw an equilateral triangle ABC of side 8 Construct a line L, such that BD: AD = L: AB. 5. Draw a line AB of length 6 cms. Divide it externally at C in the ratio 1:7. Measure AC.

Measure L.

6. Show how to find geometrically the value of a x b÷c, where a, b, and c are given numbers.

Hence evaluate geometrically (i) 29 × 17 ÷ 19; (ii) 2·1 × 3·2; (iii) 4·5 ÷ 1·7; (iv) 4·2 ÷ 7·3.

7. Find the mean proportional to the side and altitude of an equilateral triangle whose altitude is 4 cms.

8. Show how to find geometrically the value of abcd, where a, b, c, are given numbers.

Hence evaluate geometrically the fourth roots of 12, 30, and 12000. 9. Evaluate by geometrical constructions (i) √(2√2), (ii) √(10√7), (iii) √(84√69).

RIDERS.

10. Use RC. 1 to find a fourth (or third) proportional.

11. Use RC. 2 to find a fourth (or third) proportional. 12. Use RC. 3 to find a mean proportional.

13. Divide a given straight line into two parts such that their mean proportional shall be equal to a given straight line.

14 The Geometric Mean between any two quantities is less than the Arithmetic Mean, but approaches to it in value as the two quantities become more nearly equal.

15. On a given straight line construct a rectangle equal to a given rectangle.

16. From a point P outside a circle draw a line PQR meeting the circle and such that QR is a mean proportional between PQ and PR.

17. 00, OR are two intersecting straight lines and P is any point in the angle which they form. Draw through P a straight line QPR such that OP is a mean proportional between PQ and PR.

18.* OL, OM, ON are three concurrent straight lines. Prove that the locus of a point P which moves so that its perpendicular distance from OM is a mean proportional between its perpendicular distances from OL and ON is a pair of straight lines.