9. The from the right angle on the hypotenuse of a rightangled is a harmonic mean between the segments of the hypotenuse made by the point of contact of the inscribed circle.

10. If a line be cut harmonically by two Os, the locus of the foot of the 1, let fall on it from either centre, is a O, and it cuts any two positions of itself homographically (see Prop. 3, Cor. 2, Section VII.).

11. Through a given point to draw a line, cutting the sides of a given ▲ in three points, such that the anharmonic ratio of the system, consisting of the given point and the points of section, may be given.

12. If squares be described on the sides of a ▲ and their centres joined, the area of the A so formed exceeds the area of the given triangle by 3th part of the sum of the squares.

13. The locus of the centre of a O bisecting the circumferences of two fixed Os is a right line.

14. Divide a given semicircle into two parts by a to the diameter, so that the diameters of the Os described in them may be in a given ratio.

15. The side of the square inscribed in a ▲ is half the harmonic mean between the base and perpendicular.

16. The Os described on the three diagonals of a quadrilatera are coaxal.

A of

17. If X, X' be the points where the bisectors of the a A and of its supplement meet the side BC, and if Y, Y'; Z, Z', be points similarly determined on the sides CA, AB; then

18. Prove Ptolemy's Theorem, and its converse, by inversion

19. A line of given length slides between two fixed lines: find the locus of the intersection of the Ls to the fixed lines from the extremities of the sliding line, and of the 1s on the fixed lines from the extremities of the sliding line.

20. If from a variable point P 1s be drawn to three sides of a A; then, if the area of the A formed by joining the feet of these Is be given, the locus of P is a circle.

21. If a variable O touch two fixed Os, its radius varies as the square of the tangent drawn to it from either limiting point.

22. If two Os, whose centres are O, O', intersect, as in Euclid (I. 1), and 00' be joined, and produced to A, and a

GDH be described, touching the Os whose centres are O, O', and also touching the line AO; then,.

if we draw the radical axis EE' of the Os, intersecting 00' in C, and the diameter DF of the

GHD, and join

EF, the figure CDFE is a A

square.

Dem. The line joining the points of contact G and H will pass through C, the internal centre of similitude

of the Os O, O'; therefore CG. CH = CE2; but CD2 = CG. CH; therefore CD = CE.

Again, let O" be the centre of GDH, and D' the middle point of AO; then the O whose centre is D' and radius D'A touches the Os 0, 0'; hence (by Theorem 7, Section V.) the 1 from O" on EE': 0′′D :: CD': Ď′A; that is, in the ratio of 2:1. Hence the Proposition is proved.

23. If a quadrilateral be circumscribed to a O, the centre and the middle points of the diagonals are collinear.

24. If one diagonal of a quadrilateral inscribed in a O be bisected by the other, the square of the latter = half the sum of the squares of the sides.

25. If a ▲ given in species moves with its vertices on three fixed lines, it marks off proportional parts on these lines.

26. Through the point of intersection of two Os draw a line so that the sum or the difference of the squares of the chords of the Os shall be given.

27. If two Os touch at A, and BC be any chord of one touching the other; then the sum or difference of the chords AB, AC bears to the chord BC a constant ratio. Distinguish the two cases.

28. If ABC be a ▲ inscribed in a O, and if a || to AC through the pole of AB meet BC in D, then AD is = CD.

29. The centres of the four Os circumscribed about the As formed by four right lines are concyclic.

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

31. If a variable line meet four fixed lines in points whose anharmonic ratio is constant, it cuts these four lines homographically.

32. Given the CD to the diameter AB of a semicircle, it is required to draw through A a chord, cutting CD in E and the semicircle in F, such that the ratio of CE : EF may be giv n.

33. Draw in the last construction the line AEF so that the quadrilateral CEFB may be a maximum.

34. The described through the centres of the three escribed Os of a plane A, and the circumscribed of the same ▲, will have the centre of the inscribed O of the ▲ for one of their centres of similitude.

35. The Os on the diagonals of a complete quadrilateral cut orthogonally the O described about the A formed by the three diagonals.

36. When the three Is from the vertices of one ▲ on the sides of another are concurrent, the three corresponding is from the vertices of the latter, on the sides of the former, are concurrent.

37. If a O be inscribed in a quadrant of a O; and a second O be described touching the O, the quadrant, and radius of quadrant; and a be let fall from the centre of the second O on the line passing through the centres of the first and of the quadrant; then the ▲ whose angular points are the foot of the 1, the centre of the quadrant, and the centre of the second O, has its sides in arithmetical progression.

38. In the last Proposition, the Is let fall from the centre of the second on the radii of the quadrants are in the ratio of 1:7.

39. When three Os of a coaxal system touch the three sides of a ▲ at three points, which are either collinear or concurrently connectant with the opposite vertices, their three centres form, with those of the three Os of the system which pass through the vertices of the ▲, a system of six points in involution.

40. If two Os be so placed that a quadrilateral may be inscribed in one and circumscribed to the other, the diagonals of the quadrilateral intersect in one of the limiting points.

41. If from a fixed point is be let fall on two conjugate rays of a pencil in involution, the feet of the 1s are collinear with a fixed point.

42. MIQUEL'S THEOREM.-If the five sides of any pentagon ABCDE be produced, forming five As external to the pentagon,

the Os described about these As intersect in five points A", B", C", D", E", which are concyclic.

G

E"

B'

Dem.-Join E'B', E"D", D"C", C"B", C"C; join also D'D and E'B", and let them produced meet in G. Now, consider the ▲ AB'E', it is evident the O described about it (Cor. 3, Prop. 12, Book III.) will pass through the points E", B"; hence the four points E", B', E', B" are concyclic; .. the GB"E" = E'B'E"; but E'B'E" GD"E": L GB"E": GD"E". Hence the O through the points B", D", E" passes through G.

=

=

=

=

Again, since the figure CDD"C" is a quadrilateral in a O, the 4 GDE' D"C"C, and the GE'D B"C"C (III. 21); ... 4 B"C"D" = GDE' + GE'D. To each add E'GD, and we see that the figure GD"C"B" is a quadrilateral in a O; hence the through the points B", D", E" passes through C". In like manner it passes through A". Hence the five points A", B", C", D", E" are concyclic.

43. If the product of the tangents, from a variable point P to two given Os, has a given ratio to the square of the tangent from P to a third given O coaxal with the former, the locus of P is a circle of the same system,

44. Through the vertices of any A are drawn any three parallel lines, and through each vertex a line is drawn, making the same with one of the adjacent sides which the parallel makes with the other; these three lines are concurrent. Required the locus of the point in which they meet.

45. If from any point in a given line two tangents be drawn to a given ©, X, and if a O, Y, be described touching X and the two tangents, the envelope of the polar of the centre of Y with respect to X is a circle.

46. The extremities of a variable chord XY of a given O are joined to the extremities of a fixed chord AB; then, if m AX.AY +n BX. BY be given, the envelope of XY is a circle.

47. If A, A' be conjugate points of a system in involution, and if AQ, A'Q be to the lines joining A, A' to any fixed point P, it is required to find the locus of Q.

48. If a, a, b, b', c, c', be three pairs of conjugate points of a system in involution; then,

49. Construct a right-angled ▲, being given the sum of the base and hypotenuse, and the sum of the base and perpendicular. 50. Given the perimeter of a right-angled ▲ whose sides are in arithmetical progression: construct it.

51. Given a point in the side of a A; similar to a given ▲, and having one

inscribe in it another ▲ at the given point.

52. Given a point D in the base AB produced of a given ▲ ABC; draw a line EF through D cutting the sides so that the area of the A EFC may be given.

53. Construct a ▲ whose three s shall be on given Os, and whose sides shall pass through three of their centres of similitude.

54. From a given point O three lines OA, OB, OC are drawn to a given line ABC; prove that if the radii of the Os inscribed in OAB, OBC are given, the radius of the inscribed in OAC will be determined.

55. Equal portions OA, OB are taken on the sides of a given right AOB, the point A is joined to a fixed point C, and a 1 let fall on AC from B: the locus of the foot of this is a circle.