GEOMETRICAL GYMNASIUM.

1. The line AB is divided internally in O, and externally in O', in the same ratio; then OO' is a harmonic mean between AO′ and BO'.

Definition. The line AO' is said to be cut harmonically, and the points O and O' are called harmonic conjugates with respect to the points A, B.

2. If AB be bisected in C, prove that the rectangle under CO and CO' is equal to the square of AC.

3. The rectangle under the arithmetic and the harmonic means between any two lines is equal to the square of the geometric mean.

4. The reciprocals of lines in harmonic proportion are in arithmetic proportion.

5. The perpendicular of a right-angled triangle is a harmonic mean between the segments of the hypotenuse made by the point of contact of the inscribed circle.

6. Given two points A and B, find a pair of harmonic conjugates O and O' to them, such that there shall be given

1°. 00'.

2o. AO+AO'.
3°. AO2 AO2.

4°. AO × AO'.

7. If two circles intersect at right angles, then any line through the centre of either is cut harmonically by the other.

8. A right line is cut harmonically by two given circles; prove that the rectangle under the perpendiculars let fall on it from the centres of the circles is constant.

9. Find the locus of the middle point of the chords of either circle

in the last.

10. Draw a line which shall be cut harmonically by two given

circles

1o. Passing through a given point.
2o. Parallel to a given line.

11. If four right lines diverging from a point divide any line into segments in harmonic proportion, they will divide every other line also harmonically.

Definition. Such a system of four lines is called a harmonic pencil. 12. If two alternate legs of a harmonic pencil intersect at right angles, they will form the internal and external bisectors of the angle between the other legs of the pencil.

13. Through any point O, lines are drawn from the angles of a triangle ABC meeting the sides in the points A', B', and C', respectively; prove that AB': B'C AC'x BA': BC'× CA.

=

14. If any transversal be drawn cutting the sides of a triangle in the points a, b, c, respectively, prove that Ab : bC = Ac × Ba: Bc × Ca.

15. If the lines joining the points A' and B', in question (13), be produced to meet the base AB in a point c, prove that the side AB is cut harmonically at the points C' and o.

16. In the same figure, if B'C' and A'C' be also joined and produced to meet the opposite sides in the points a and b respectively, prove that the points a, b, c are situated in one right line.

17. Given one pair of the opposite sides of a quadrilateral, and the point of intersection of the other pair produced to meet; find the locus of the intersection of its diagonals.

18. If the external bisectors of the angles of a triangle be produced to meet the opposite sides, the three points of intersection will be situated in a right line.

19. The right lines drawn from the angles of any triangle to the points of contact of the inscribed circle pass through the same point.

20. Given a perpendicular to the diameter of a given semicircle; it is required to draw from one extremity of its diameter a chord cutting the perpendicular and semicircle, such that, joining the opposite extremity of the diameter with its point of intersection with the semicircle, the quadrilateral so formed shall be a maximum.

21. If A, B, C, D be four points taken in order on a right line; prove that AC × BD = AB × CD + AD × BC.

22. If the right line be divided harmonically in these four points, prove that AC x BD = 2AB × CD.

23. Any right line is intersected by four lines diverging from the same point in A, B, C, D; prove that AD × BC is to AB × CD in a constant ratio: as also AC × BD to AB × CD (Prop. K).

Definition. These constant ratios are called the anharmonic ratios of the pencil, and also of the four points.

24. If two pencils have one pair of their anharmonic ratios equal, their other anharmonic ratios will also be equal.

25. If the line joining the vertices of two pencils be a ray common to both, and they have the same anharmonic ratio, the intersection of their corresponding rays will lie in directum.

26. The sides of a triangle pass through three points in directum, and two of its angles move on given lines; find the locus of the remaining angle.

27. The anharmonic ratio of four points A, B, C, D, situated on the circumference of a circle, is the ratio of the rectangle under AB and CD to that under AC and BD.

Definition. The anharmonic ratio of four points on a circle is that of the pencil formed by joining them to any fifth point on the circle.

28. If the opposite sides of a hexagon inscribed in a circle be produced to meet, their three points of intersection lie in a right line.

29. The tangents at the angular points of any triangle inscribed in a circle intersect the opposite sides in three points which are situated in a right line.

30. Given two systems of points, A, B, C, A', B', C', on a circle; find a point P, such that the anharmonic ratio of P, A, B, C, and of P, A', B', C', shall be equal, and show that the problem admits of two solutions.

31. If in the last problem a right line be substituted for the circle show how the point P can be determined.

32. The bases of two triangles inscribed in a circle are fixed; prove that the line joining the points of intersection of their corresponding sides passes through a fixed point.

33. Inscribe in a given triangle another, each of whose sides shall pass through a fixed point.

34. Through a given point draw two transversals which shall intercept given lengths on two given lines.

35. Describe a triangle of given vertical angle; its sides passing through three given points; and its base angles resting on given lines.

36. A chord is drawn through a fixed point either inside or outside a circle, and tangents at its extremities; find the locus of their intersection.

Definition. The point and the locus of the right line are said to be a pole and polar with respect to the given circle.

37. Any right line drawn through the pole is cut harmonically by the circle and the polar.

38. If any number of points lie in a right line, their polars, with respect to any circle, all pass through the same point.

39. The right line joining any two points has for its pole the intersection of the polars of the given points.

40. If four points lie in directum, their anharmonic ratio is the same as that of the pencil formed by their four polars.

41. If through any point two right lines be drawn cutting a circle, the lines joining the points of intersection will meet on the polar of the fixed point.

42. If through any point four right lines be drawn cutting a given circle, the anharmonic ratio of any four of the points of section is equal to that of the remaining four.

43. If two tangents and a secant be drawn from any point outside a circle, the two points of contact and the points of section will subtend a harmonic pencil at any point on the circle.

44. If two points on a circle be fixed, and any other pair of points taken dividing the circle harmonically, the chord joining these points will always pass through a fixed point.

45. Draw a transversal passing through a given point and cutting a given arc harmonically.

Definition.-An arc is divided harmonically when the two points of section and the extremities of the arc subtend a harmonic pencil at any fifth point on the circle.

46. Draw a right line intersecting two given arcs harmonically.

47. If a variable circle divide two given arcs of two circles harmonically, the locus of its centre is a right line.

48. Describe a circle dividing three given arcs of three circles harmonically.

49. Describe a circle passing through two given points and dividing an arc of a given circle harmonically.

50. If a quadrilateral be inscribed in a circle, and another circumscribed touching at the angular points, prove the following —

1o. Their diagonals intersect in the same point.

2°. Their third diagonals are coincident.

3°. The intersection of each pair of their three diagonals is the pole of the remaining one.

51. The lines joining the opposite angles of a hexagon circumscribed to a circle pass through the same point.

52. If OA and OB be two fixed chords of a circle; OX and OY two variable chords which form an harmonic pencil with OA and OB, the chord XY passes through a fixed point.

53. If two tangents be drawn to a circle, any third tangent will be cut harmonically by the two former, and by the chord joining their points of contact.

54. Three of the angles of a quadrilateral, whose angles are given, move on three right lines given in position; find the locus of its fourth angle.

55. If a quadrilateral be inscribed in a circle, such that three of its sides pass through given points on a right line, the fourth side will pass through a fourth fixed point situated on the same right line.

56. If two triangles be such that the vertex of each is the pole of a side of the other, then will the three points of intersection of the corresponding sides lie in a right line, and the lines joining the corresponding angles pass through one point.

57. Given two pairs of points X, X'; Y, Y', on a right line; a point O can always be determined such that OX × OX' shall be equal to OY × OY'.

58. If XY = X'Y', the point O is situated at an infinite distance. 59. If two other points Z, Z' be taken on the line, such that OZ × OZ'= 0XX OX' = OY OY', then the system of points X, X, Y, Y', and Z, Z', are such that the anharmonic ratio of any four of them is equal to that of their four conjugates.

Definition. Each pair of such points X, X, Y, Y', or Z, Z' are called conjugates; and the system of six points is said to be in involution; O is called the centre of the system; and two points F, F' such that OF2 = OF′2 = OX × OX' are called the foci or double points of the system.

60. In what case are the foci F and F real points, and when imaginary?

61. There is an indefinite number of pairs of points in involution, with two given pairs of points on a right line.

62. Find the locus of a point at which two given portions of the same line subtend equal angles.

63. A right line meeting the sides and diagonals of any quadrilateral is divided in six points in involution.

64. Any right line meeting a circle and the sides of any inscribed quadrilateral is cut in involution.

65. The lines drawn from any point to the six angular points of a complete quadrilateral form a system in involution.

66. If three circles have the same radical axis, then any transversal will intersect them in a system of points in involution.

67. How can the centre and foci of the system be determined?