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13. If AB be divided in medial section internally at H and externally at K, prove that: (i) The rectangle BK, HA is equal to the rectangle BA, AK.

(ii) The rectangle BH, AK is equal to the rectangle BA, HA.
(iii) The square on HK is equal to five times the square on AB.
(iv) The sum of the squares on AH and AK is equal to three

times the square on AB.
(v) The sum of the squares on BH and BK is equal to seven

times the square on AB. 14. If AB be divided internally at C and externally at D so that the square on AC is double the square on BC, and the square on AD is double the square on BD, prove that :

(i) the rectangle AC, AD is equal to twice the square on AB. (ii) the rectangle CB, BD is equal to the square on AB. (iii) the rectangle AD, CB is equal to the rectangle AC, BD. (iv) the square on CD is equal to eight times the square on AB. (v) if P be the middle point of CD, B is the middle point of AP. (vi) the difference between the rectangle AD, BD and the rectangle AC, CB

is equal to eight times the square on AB. 15. If AB be divided internally at P and externally at ( so that the square on AP is equal to twice the rectangle AB, PB, and the square on AQ is equal to twice the rectangle AB, QB, prove that :

(i) the rectangle PA, AQ is equal to twice the square on AB. (ii) the rectangle BP, BQ is equal to the square on AB. (iii) the rectangle PA, BQ is equal to the rectangle BA, AQ. (iv) the rectangle BP, AQ is equal to the rectangle BA, PA. (v) the sum of the squares on AP, AQ is equal to eight times the square

on AB. (vi) if M be the middle point of PQ, A will be the middle point of BQ.

On Propositions 12 and 13. 16. In any triangle three times the sum of the squares on the sides of the triangle is equal to four times the sum of the squares on the medians.

17. ABC is a triangle having the angle BAC a right angle; prove that four times the sum of the squares on the medians BE and CF is equal to five times the square on the hypotenuse BC.

18. ABC is any triangle and its medians intersect in G; if P be any point, show that the sum of the squares on AP, BP, CP is greater than the sum of the squares on AG, BG and CG by three times the square on GP.

19. ABC is an isosceles triangle having the side AB equal to the side AC; show that if AB is produced to D so that AB equals BD, the square on CD is equal to the square on AC, together with twice the square on BC,

20. If DE be drawn parallel to the base BC of an isosceles triangle ABC, the square on BE is equal to the square on CE, together with the rectangle contained by BC, DE.

21. The rectangle contained by the sum and difference of any two sides of a triangle is equal to twice the rectangle contained by the base and the line intercepted between the middle point of the base and the foot of the perpendicular from the vertex to the base.

22. ABC is an acute-angled triangle, and BE and CF are drawn perpendicular to the opposite sides ; prove that the sum of the rectangles AB, FB and AC, EC is equal to the square on BC.

23. ABC is an acute-angled triangle, and perpendiculars APD, BPE are drawn on BC, CA from the opposite angles; prove that the rectangle AD, PD is equal to the rectangle BD, DC.

24. In the figure of Prop. 13 if BE be drawn perpendicular to AC, prove that the rectangle AC, EC is equal to the rectangle BC, DC.

25. ABC is an acute-angled triangle and perpendiculars APD, BPE, CPF are drawn to the three sides from the opposite angles; prove that the rectangles AP, PD and BP, PE and CP, PF are all equal to one another.

26. ABC is an isosceles triangle having each of the angles at the base double the vertical angle BAC; if CD be drawn perpendicular to AB, prove that :

(i) the square on AC is equal to twice the rectangle AD, BC; (ii) the sum of the squares on AD, DB is equal to the square on CD; (iii) the square on AC is equal to the square on BC, together with the

rectangle AC, BC; (iv) the rectangle BA, BD is equal to half the square on BC; (v) the square on CD is to the square on BD, together with twice

the rectangle BD, DA. 27. If AD be drawn perpendicular to BC, a side of the triangle ABC, prove that the angle BAC will be obtuse or acute according as the rectangle BD, DC is greater or less than the square on AD.

28. ABC is any triangle, having B and C acute angles, and squares are described externally on its three sides, using the same lettering as in I. 47 ; then BN and CM are drawn perpendicular to CA and BA respectively, produced if necessary; complete the rectangles HANR and GAMQ and prove them equal to each other.

Hence deduce proofs of Props. 12 and 13, according as the angle BAC is obtuse or acute.

29. In the figure of I. 47, if BAC be any angle, prove that the difference of the squares on AB and AC is equal to the difference of the squares on AD and AE.

30. The straight line AB is divided at C so that AC is double CB, and P is any point; prove that the square on PA, together with twice the square on PB, is equal to three times the square on PC, together with six times the

square on CB,

31. ABCD is a parallelogram having the angle BAD 60°; show that the square on AC is equal to the sum of the squares on AB, BC together with the rectangle AB, BC.

32. In any qnadrilateral the sum of the squares on the diagonals is equal to twice the sum of the squares on the straight lines joining the middle points of the opposite sides.

33. In any quadrilateral, two of whose opposite sides are parallel, the sum of the squares on the diagonals is equal to the sum of the squares on its nonparallel sides, together with twice the rectangle contained by the parallel sides.

34. If the sum of the squares on the sides of the quadrilateral be equal to the sum of the squares on its diagonals, it must be a parallelogram.

35. If the diagonals of a parallelogram intersect in G, and P be any point,

prove that

PA? + PB 2 + PC 2 + PD ? = 4 PG? + AB 2 + AD 2. 36. If ABCD be a parallelogram, and P any point, and if the sum of the squares on PA and PC be equal to the sum of the squares on PB and PD, prove that ABCD is rectangular.

37. If A, B, C, D are fixed points, and P is a point such that the sum of the squares on PA, PB, PC, PD is coustant, prove that the locus of P is a circle, whose centre is the point of intersection of the straight lines joining the middle points of AB, CD and of AD, BC.

38. If A, B, C, D are fixed points, and P is a point such that the sum of the squares on PA, PB is equal to the sum of the squares on PC, PD, prove that the locus of P is a straight line at right angles to the line joining the middle points of AB, CD.

39. In the last question prove that the locus of P passes through the intersection of the lines drawn perpendicular to either of the pairs of lines AC, BD or AD, BC at the middle points.

PROBLEMS. 40. Divide a given straight line into two parts so that the rectangle contained by the parts may be a maxinium.

41. Divide a given straight line into two parts so that the sum of the squares on the two parts may be a minimum.

42. Divide a given straight line into three parts, so that the sum of the squares on the three parts may be a minimum.

43. Divide a given straight line into two parts so that the difference of the squares on the parts may be equal to a given rectilineal figure. When is this impossible?

44. Divide the straight line AB at the points C and D, so that, if AC be equal to DB, the square on AC may be equal to the rectangle AB, CD.

45. Divide a given straight line into two parts so that the sum of the squares on the whole and one part may be equal to (i) twice, (ii) three times, (iii) four times, or (iv) five times the square on the other part.

46. Find the length of the side of a square equal in area to a rhombus whose side is three inches long and one of whose angles is 150°.

47. Divide a straight line into two parts so that the rectangle contained by the two parts may be equal to a given square. Is this always possible ?

48. Describe a rectangle equal to a given rectilineal figure and having its perimeter a minimum.

49. Describe a rectangle equal to a given square and having the difference of two of its adjacent sides equal to a given straight line.

50. Describe a square equal to a given regular pentagon.

51. Describe a square whose area shall be equal to the sum of the areas of the squares described on the three sides of a given right-angled triangle.

52. Describe a right-angled triangle having one of its sides double of the other and equal in area to a given rectilineal figure.

53. Describe a parallelogram equal in area to a given square, having its perimeter equal to a given straight line, and having an angle equal to a given angle. When is this impossible ?

54. Find the locus of a point, such that the sum of the squares on its distances from three given fixed points is constant.

55. ABC is any triangle; find a point p within it such that the sum of the squares on AP, BP and CP is a minimum.

56. Find the position of a point such that the sum of the squares on its distances from four given fixed points is a minimum.

57. Given the base of a triangle and the sum of the squares on its sides, find the locus of its vertex.

PRACTICAL EXERCISES ON BOOK II.

2

7 cm.,

1. Show that Prop. 7 is equivalent to the algebraic formula

(a - b)2 = a2 +62 – 2ab. 2. Show that Prop. 8 is equivalent to the algebraic formula

(a+b)2 – (a - b)2 = 4ab. 3. In the figure of Prop. 14 it follows from Euclid's arguments that EH2 BE. EF. Use this figure to construct a line of length V14 cm. and measure the line. [Make BE

EF 2 cm.] [Note that we need not construct the rectangle BD.]

4. Construct a line of length V3.3" and measure it. [Make BE = 3", EF 1.1".]

5. Using the figure of Prop. 14 construct and measure lines of the following lengths :-(i) V7", (ii) v10 cm., (iii) V19 cm. [19 = 5x3.8], (iv) V23 cm., (v) V4.5 in., (vi) V3:4 in.

6. Describe a rectangle whose perimeter is inches and whose area is 1.5 square inches. Measure its sides. [Using the figure of Prop. 14:-Make BF = 3"; describe the semicircle ; make FK V1:5" and I to BF; draw KH || to BF, meeting the circumference in H; draw HE I to BF; construct the rectangle BEDC.]

7. Describe a rectangle whose area is 9 sq. cm. and whose perimeter 14 cm.

Measure its sides. 8. Find geometrically two numbers whose sum is 7 and whose product is 9. [Cf. question 7.]

9. Find geometrically two numbers whose sum is 20 and whose product 70. [Work in tenth-inches.]

10. Show how to produce a given line to such a length that the rectangle contained by the whole line and the part produced shall be equal to the square on a given line. [Use II. 6 and I. 47.] Hence find geometrically two numbers whose difference is .4 and whose product is 20.

11. Use Add. Prop. 5, p. 148, to calculate the length of the median of a triangle if the base is 3 inches, and the sum of the squares of the sides is 12 square inches.

12. Draw the locus of the vertex of a triangle on a given base of length 3 inches, if the sum of the squares on the sides is 12 square inches. [Cf. question 11].

13. Construct a triangle in which the base is 40 mm., the area is 400 sq. mm., and the sum of the squares on the sides is 2000 sq. mm. Measure the shorter side.

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