DEF. 5.-If the straight line joining the centres of two circles be divided internally and externally in the ratio of the radii, the points of section are called the internal and external centres of similitude of the two circles. (The phrase 'centre of similitude' is due to Euler, 1777. See Nov. Act. Petrop., ix. 154.)

DEF. 6.-The figure which results from producing all the sides of any ordinary quadrilateral till they intersect is called a complete quadrilateral; and the straight line joining the intersections of pairs of opposite sides is called the third diagonal. (Carnot, Essai sur la Théorie des Transversales, p. 69.)

To the notation adopted for points and lines connected with the triangle ABC on pp. 98-100, 252, 253, should be added the following:

N, P, Q denote the points where the bisectors of the interior ▲ 8 A, B, C meet the opposite sides.

N', P', Q' denote the points where the bisectors of the exterior 28 A, B, C meet the opposite sides.

A by itself denotes the area of ▲ ABC.

p denotes the radius of the circle inscribed in the orthocentric Δ ΧΥΖ.

DEDUCTIONS.

1. C and D are two points both in AB, or both in AB produced: show that AC: CB is not AD: DB.

=

2. Find the geometric mean between the greatest and the least straight lines that can be drawn to the Oce of a circle from a point (1) within, (2) without the circle.

3. In the figure to IV. 10, As ABD, ACD, DCB are in geometrical progression.

4. Construct a right-angled triangle whose sides shall be in geometrical progression.

5. If a straight line be a common tangent to two circles which touch each other externally, that part of the tangent between the points of contact is a geometric mean between the diameters of the circles.

6. Any regular polygon inscribed in a circle is a geometric mean between the inscribed and circumscribed regular polygons of half the number of sides.

=

7. To find a mean proportional between AB and BC, C being situated between A and B. Produce AB to E, making BE AC; with A and E as centres and AB as` radius, describe arcs cutting in D; join BD. BD is the mean proportional. (See Wallis's Algebra, Additions and Emendations, 1685, p. 164.)

Of three straight lines in geometrical progression :

8. Given the mean and the sum of the extremes, to find the

extremes.

9. Given the mean and the difference of the extremes, to find the

extremes.

10. Given one extreme and the sum of the mean and the other

extreme, to find the mean and the other extreme.

11. Given one extreme and the difference of the mean and the other extreme, to find the mean and the other extreme.

12. Find two straight lines from any two of the six following data : their sum, their difference, the sum of their squares, the difference of their squares, their rectangle, their ratio.

13. If two triangles have two angles supplementary and other two angles equal, the sides about their third angles are propor tional.

14. Divide a straight line into two parts, the squares on which shall have a given ratio.

15. Describe a square which shall have a given ratio to a given

polygon.

16. Cut off from a given triangle another similar to it, and in a given ratio to it.

17. Cut off from a given angle a triangle = a given space, and such that the sides about that angle shall have a given ratio.

18. ACB is a semicircle whose diameter is AB, and on AB is described a rectangle ADEB, whose altitude = the chord of half the semicircle; from C, any point in the Oce, CD, CE are drawn cutting AB at F and G. Prove AG2 + BF2 = AB2. (Due to Fermat, 1658. See Wallis's Opera Mathematica, 1695, vol. i. p. 858.)

19. If two chords AB, CD intersect each other at a point E inside a circle, the straight lines AD, BC cut off equal segments from the chord which passes through E and is there bisected.

20. Enunciate and prove the preceding theorem when the chords AB, CD intersect each other outside the circle.

=

25. sar1(2 + 13), 8b = r2 (T3 + T1), SC = r3 (7'1 + 72). 26. r(r1 + r1⁄2 + r3) = AF · FB ÷ BD · DC + CE · EA. 27. A = R (XY + YZ + ZX).

28. 28: XY + YZ + ZX = R : r.

29. ▲ ABC: ▲ XYZ = R : p.

30. 2Rp 40.0X

=

31. a2 + b2 + c2 =

=

BO . OY = CO . OZ.

8R2 + 4Rp.

33. SI1 = R(R + 2r1), SI‚2 = R(R + 2r2), SI32 = R(R + 2r3). 34. SI2 + SI22 + SI22 + SI32 = 12R2.

35. a2 + b2 + c2 + p2 + r12 + r22 + r22 = 16R2.

36. II22 + II2 + II32 + I1I„2 + I2I32 + I3112 = 48R2.

[Regarding theorem 21, see p. 145. It has, however, been conjectured, and with probability, that the treatise in which it occurs is a work of Heron the younger, and therefore long subsequent to the date of the elder Heron. The theorem was known to Brahmegupta, 628 A.D. For theorems 22, 36, 25, 26, see Davies in Ladies' Diary, 1835, pp. 56, 59; 1836, p. 50; and Philosophical Magazine for June 1827, p. 28. For 23 and 24, see Lhuilier, Élémens d'Analyse, p. 224. For 27, 28, 29, 30, 31, 34, 35, see Feuerbach, Eigenschaften, &c., section vi., theorems 3, 4, 5, 6, 7; section iv., § 50; section ii., § 29. Theorem 32 is usually attributed to Euler, who gave it in 1765. It occurs, however, in vol. i., art. xvii., by William Chapple, of the Miscellanea Curiosa Mathematica, and probably appeared about 1746. Theorem 33 is given in John Landen's Mathematical Lucubrations, 1755, p. 8. Some of the properties 37-40 are well known; but I cannot trace them to their sources. Hundreds of other beautiful properties of the triangle may be found in Thomas Weddle's papers in the Lady's and Gentleman's Diary for 1843, 1845, 1848.]

Construct a triangle, having given :

41. The vertical angle, the ratio of the sides containing it, and the base. (Pappus, VII. 155.)

42. The vertical angle, the ratio of the sides containing it, and the diameter of the circumscribed circle.

43. The vertical angle, the median from it, and the angle which the median makes with the base.

44. The vertical angle, the perpendicular from it to the base, and the ratio of the segments of the base made by the perpendicular.

45. The vertical angle, the perpendicular from it to the base, and the sum or difference of the other two sides.

46. The base, the perpendicular from the vertex to the base, and the ratio of the other two sides.

47. The base, the perpendicular from the vertex to the base, and the rectangle contained by the other two sides.

48. The segments into which the perpendicular from the vertex divides the base, and the ratio of the other two sides.

49. The perpendiculars from the vertices to the opposite sides. 50. The sides containing the vertical angle, and the distance of the vertex from the centre of the inscribed circle.

TRANSVERSALS.

The following five triads of straight lines are concurrent :

1. The medians of a triangle.

2. The bisectors of the angles of a triangle.

3. The bisector of any angle of a triangle and the bisectors of the two exterior opposite angles.

4. The perpendiculars from the vertices of a triangle on the opposite sides.

5. AL, BK, CF in the figure to I. 47.

6. If two sides of a triangle be cut proportionally (as in VI. 2), the straight lines drawn from the points of section to the opposite vertices will intersect on the median from the third vertex; and conversely.

7. The points in which the bisectors of any two angles of a triangle and the bisector of the exterior third angle cut the opposite sides are collinear.

8. The points in which the bisectors of the three exterior angles of a triangle meet the opposite sides are collinear.

9. If a circle be circumscribed about a triangle, the points in which tangents at the vertices meet the opposite sides are collinear. 10. The perpendiculars to the bisectors of the angles of a triangle

at their middle points meet the sides opposite those angles in three points which are collinear. (G. de Longchamps.) 11. OA, O'A', O"B" are three parallel straight lines; 00', AA' meet at B"; O'O", A'A" at B; 0"O, A"A at B'. Prove B, B', B" collinear.

12. If a transversal cut the sides, or the sides produced, of any polygon, the product of one set of alternate segments taken cyclically is equal to the product of the other set. (Carnot's Essai sur la Théorie des Transversales, p. 70.)

13. If a hexagon be inscribed in a circle, and the opposite sides be produced to meet, the three points of intersection are collinear. (Particular case of Pascal's theorem.)

14. Prove with reference to fig. on p. 345.

AO BO CO: DO · EO · FO = AB · BC. CA: AF. BD.CE.

(Davies's edition of Hutton's Mathematics, 1843, vol. ii. p. 219.)

15. If a point A be joined with three collinear points B, C, D, then will

AC2 · BD ± AB2 · CD = AD2 · BC ± BD · DC • BC, the upper sign being taken when D lies between B and C, and the lower when it does not. (Matthew Stewart's Some General Theorems of considerable use in the higher parts of Mathematics, 1746, Prop. II.) Deduce from the preceding theorem, App. II. 1; deduction 1 on p. 151; VI. B; and App. VI. 8.

16. If the Oce of a circle cut the sides BC, CA, AB, or those sides produced, of ▲ ABC at the points D,D', E‚E', F,F', then will AF AF BD. BD'.CE. CE' = FB.F'B.DC. D'C.EA.E'A. (Carnot's Essai, &c., p. 72.)

17. Prove with reference to fig. on p. 251.

AIBI CI: AB. BC. CA = AB. BC CA: AI1· BI2 · CI3.

(C. Adams's Die merkwürdigsten Eigenschaften des geradlinigen Dreiecks, 1846, p. 20.)

18. Prove the following triads of straight lines connected with A ABC concurrent: