22. In the figure of Prop. 7, if FG, KH be produced to meet in P, prove that PA produced cuts BC at right angles. 23. Points E, F, G, HI are taken respectively in the sides. AB, BC, CD, DA of a rectangle ABCD; if EF = GH, prove that

AG2 + CH2 = AF2 + CE2.

24. From D, the middle point of the side AC of a right triangle ABC, DE is drawn perpendicular to the hypotenuse AB: prove that BE AE + BC'.

=

25. ABCD is a parallelogram. If AC is bisected in O and a straight line MON is drawn to meet AB, CD in M, N respectively, and OR parallel to AB meets AN in R; prove that the triangles ARM, CRN are equal.

NC AM (105), .. AN is | to CM (133), etc.

26. If through the vertices of a triangle ABC there be drawn three parallel straight lines AD, BE, CF to meet the opposite side or sides produced in D, E, F: prove that the area of the triangle DEF is double that of ABC.

Let D be in BC; E, F in CA, BA produced; ▲ EFB = ▲ ECB, etc.

27. ABC, DEF are triangles having the angles A and D equal, and AB equal to DE: show that the triangles are to each other as AC to DF.

28. The side BC of the triangle ABC is bisected in D: prove that any straight line through D divides the sides AB, AC into segments which are proportional.

29. On the sides AB, AC of a triangle ABC points D, E are taken such that AD is to DB as CE is to EA: if the lines CD, BE intersect in F, prove that the triangle BFC is equal to the quadrilateral ADFE.

30. ABCD is a square. A line drawn through A cuts the sides BC, CD, produced if necessary, in E, F. Prove that the triangle CEF is to the triangle ABC as the difference of the lines CE and CF is to BC.

31. O is a point inside a triangle ABC; AO, BO, CO produced meet the sides in D, E, F, respectively. If AO is to OD as BO to EO as CO to OF, prove that O is the centroid (172) of the triangle ABC.

32. ABC is a triangle, and points D, E are taken in BC, CA respectively, such that BD = BC and AE = {AC; AD, BE intersect in O. Prove that ▲ AOB = † ^ ABC.

▲ BAD: ▲ ADC = BD: DC=1: 2; sim, ▲ OBD: ▲ ODC= 1: 2, etc.

33. ABCD is a quadrilateral having two of its sides AB, CD parallel; AF, CG are drawn parallel to each other meeting BC, AD, respectively, in F, G. Prove that BG is parallel to DF.

Produce CB, DA to meet in E, etc.

34. ABC is a triangle, G its centroid, D the middle point of BC, AE its perpendicular to BC. Show that if the rectangle of which AE, EC are adjacent sides be completed, the fourth corner being F, then FG produced bisects BE.

Produce FG to meet BC in H; HD: AF = 1:2, etc.

35. ABCD, AB'C'D' are two squares, BAB', DAD' being straight lines; B'C meets AD in E, and C'D meets AB' in F. Prove that AE AF.

36. A triangle ABC has the side AB = 2AC; from C is drawn CD to a point D in AB such that ACD = /ABC: show that ▲ BCD = 3 ▲ ACD.

37. If in an isosceles triangle ABC the two equal sides AB, CA be divided at F and E respectively, in any given ratio; the straight line FE will meet BC produced at a point D, such that CD : BD = AF: FB2, or CE: EA'

Draw AG || to CB meeting EF produced in G, etc.

=

38. ABC is an isosceles triangle having the sides AB, AC equal, and is such that if BD be drawn bisecting the angle ABC and cutting AC in D, then AD is equal to BC. Show that the angle C is double the angle A.

39. Perpendiculars are drawn from the vertices of a triangle on any straight line through the centroid of the triangle prove that one of these perpendiculars is equal to the sum of the other two.

40. Perpendiculars are drawn from the vertices and the centroid of a triangle on a given straight line: prove that the perpendicular from the centroid is the arithmetic mean of the perpendiculars from the vertices.

Let ABC be the A, O its centroid, D the mid. pt. of BC; draw AL, BM, CN, DR, OK to given line; bisect AO in E; draw EH 1 to LM, etc.

41. If the vertical angle C of a triangle ABC be bisected by a line which meets the base in D, and is produced to a point E, so that the rectangle of CD and CE is equal to that of AC and CB; show that if the base and vertical angle be constant the position of E is fixed.

Since CD × CE = AC × CB, . '. CD: AC = BC: CE, and ▲ ACD = 2BCD,... ¿CAB = CEB, etc.

42. ABC is a triangle inscribed in a circle; from A straight lines AD, AE are drawn parallel to the tangents at B, C respectively, meeting BC, produced if necessary, in D, E: prove that BD is to CE as the square on AB is to the square on AC.

Let BF, CG be the tangs.: ZADB = alt. ≤DBF = <BAC (238 and 243), and B is common to both as BAC, BAD, etc.

43. If P be a point on a diameter AB of a circle, and PT be the perpendicular on the tangent at a point Q; show that PT × AB = AP × PB + PQ2.

Produce QP to meet the in R; draw diam. RS; TQP = <QSR, etc. 44. Prove that the square constructed

on the sum of two straight lines is equivalent to the sum of the squares constructed on the two lines, together with twice the rectangle of the lines.

45. Prove that the square constructed on the difference of two straight lines is equivalent to the sum of the squares constructed on the lines, diminished by G twice the rectangle of the lines.

46. Prove that the rectangle of the

D G

E

H

F

K

CB

ск

B

H

G

sum and the difference of two straight

lines is equivalent to the difference of D' E the squares constructed on the lines.

47. ABC is an isosceles triangle, CA, CB being the equal sides; BO is drawn at right angles to BC to meet CA produced in O: show that the square on OB is equal to the square on OA with twice the rectangle OA, AC.

48. If a straight line be divided into two equal and

also into two unequal parts, the rectangle of the unequal parts with the square on the line between the points of section, is equal to the square on half the line. (Euclid, B. II, prop. 5).

49. The square on the straight line, drawn from the vertex of an isosceles triangle to any point in the base, is less than the square on a side of the triangle by the rectangle of the segments of the base.

50. The sides AB, CD of a quadrilateral ABCD inscribed in a circle are produced to meet in P; and PE, PF are drawn perpendicular to AD, BC respectively: prove that AE is to ED as CF to FB.

51. The sides AB, AD, produced if necessary, of a parallelogram ABCD, meet a line through C in E, F respectively; CB, CD, produced if necessary, meet a line through A in G, H respectively. Prove that GE is parallel to HF. PROBLEMS.

52. Construct a square equal to three given squares. 53. Construct a square which shall be five times as great as a given square.

54. Construct a triangle equal in area to a given quadrilateral.

55. Construct an isosceles triangle equal in area to a given triangle and having a given vertical angle.

56. Divide a straight line into two parts, so that the sum of the squares on the parts may be equal to a given square. 57. Trisect a triangle by straight lines drawn from a given point in one of its sides.

58. Find a point O inside a triangle ABC such that the triangles OAB, OBC, OCA are equal.

59. Construct a square that shall be one-third of a given

square.

60. Divide a straight line into two parts so that the rectangle contained by the whole and one part may be equal to a given square.

Let AB be the given line. Draw BD to AB so that BD = given square: join AD, draw DE 1 to AD, meeting AB produced in E.

61. Produce AB to C so that the rectangle of AB and AC may be equal to a given square.

62. Construct a parallelogram equal to a given triangle and having one of its angles equal to a given angle.

63. Given three similar triangles: construct another similar triangle and equivalent to their sum. (377).

64. Construct a triangle similar to a given triangle ABC which shall be to ABC in the ratio of AB to BC.

Find DE a mean proportional between AB, BC, etc.

65. Bisect a triangle by a straight line parallel to one of its sides.

Bisect AC of the ▲ ABC in D; from AC cut off AE a mean proportional to AC, AD, etc.

66. A is a point on a given circle: draw through A a straight line PAQ meeting the circle in P and a given straight line in Q, so that the ratio of PA to AQ may be equal to a given ratio.

Draw AB to any pt. B in the given line MN: produce BA to C so that BA: AC given ratio, etc.

67. From the vertex of a triangle draw a line to the base, so that it may be a mean proportional between the segments of the base.

About the given ▲ describe a O, etc.

68. Show how to draw through a given point in a side of a triangle a straight line dividing the triangle in a given ratio.

69. Construct a triangle equal to a given triangle and having one of its angles equal to an angle of the triangle, and the sides containing this angle in a given ratio.

70. Draw through a given point a straight line, so that the part of it intercepted between a given straight line and a given circle may be divided at the given point in a given ratio.

Let P be the given pt., XY the given line, A the cent. of given : produce AP to meet XY in Q; in PA take B so that QP: PB = given ratio, etc.