Multiply the equations (1), (2), (3) together, and strike out the common factors; then AN BL. CM = NB. LC. MA; .. L, M, N are collinear. App. VI. 1 The converse of the theorem (which may be proved indirectly) is, If two triangles be situated so that the points of intersection of corresponding sides are collinear, the straight lines joining corresponding vertices are concurrent. HARMONICAL PROGRESSION. DEF. 2.-If a straight line be cut internally and externally in the same ratio it is said to be cut harmonically; and the two points of section are said to form with the ends of the straight line a harmonic range. Thus, if AB be cut internally at C, and externally at D, in the same ratio, AB is said to be cut harmonically; and the points A, C, B, D are said to form a harmonic range. DEF. 3.-The points C and D are said to be harmonically conjugate to each other (harmonic conjugates) with respect to the points A and B. The segments AB, CD are sometimes (Chasles' Géométrie Supérieure, § 58) called harmonic conjugates. Since a straight line can be cut internally, and therefore externally in any ratio, it may be cut harmonically in an infinite number of ways. The ancient Greek mathematicians* defined three magnitudes to be in harmonical progression when the first is to the third as the difference between the first and second is to the difference between the second and third. Now, if AB be cut internally at C and externally at D in the same ratio, Hence, if AD, AB, AC be regarded as the three magnitudes, it will be seen that they are in harmonical progression, since they conform to the definition. * Pythagoras probably first. On the different progressions, see Pappus, III., section 12, W PROPOSITION 4. If C and D are harmonic conjugates with respect to A and B, then A and B are harmonic conjugates with respect to C and D. Since C and D are harmonic conjugates with respect to A and B, .. AB is cut internally at C and externally at D in the same ratio; AD: DB = AC: CB; App. VI. Def. 3 DB: CB, by alternation, V. 16 AD: AC = that is, CD is cut externally at A and internally at B in the same ratio; .. A and B are harmonic conjugates with respect to C and D. COR. 1. Hence, if A, C, B, D form a harmonic range, not only are AD, AB, AC in harmonic progression, but also AD, CD, BD. COR. 2.—The points which are harmonic conjugates to two given points are always situated on the same side of the middle of the line joining the two given points. Suppose A and B the given points, O the middle of AB. Since C and D are harmonic conjugates with respect to A and B, .. AD: DB = AC: CB. App. VI. Def. 3 Now if D be situated (as in figs. 1 and 2) to the right of O, then AD must be greater than DB; .. AC must be greater than CB, that is, C also is situated to the right of O. If D be situated (as in figs. 3 and 4) to the left of O, then AD must be less than DB; .. AC must be less than CB, that is, C also is situated to the left of O, COR. 3.-If any three of the points forming a harmonic range be given, the fourth may be determined. Four cases are all that can arise, namely, when A, C, B, or D is to be found. (1) If A, C, B are given, D can be found by dividing AB externally in the ratio AC: CB. (2) If C, B, D are given, A can be found by dividing DC externally in the ratio DB: BC. (3) If A, B, D are given, C can be found by dividing AB internally in the ratio AD: DB. (4) If A, C, D are given, B can be found by dividing DC internally in the ratio DA: AC. PROPOSITION 5. If AD, AB, AC are in harmonical progression, and the mean AB is bisected at O, then OD, OB, OC are in geometrical progression; and conversely.* Since AD, AB, AC are in harmonical progression, . OD + OB: OD – OB = OB + OC : OB – OC; OD: OB D App. VI. Def. 2 Converse of V. D COR. 1.-Since OD: OB = OB: OC, .. OB2 = OC · OD. VI. 17 Now if A and B are fixed points, OB2 is constant ; .. OC. OD is constant. Hence if OC diminishes, OD increases, that is, if C moves nearer to O, D moves farther away; and if OC increases, OD diminishes, that is, if C moves away from O, D moves nearer to 0. In other words, if C and D move in such a manner as always to remain harmonic conjugates with respect to the fixed points A and B, they must move in opposite directions. Also, the nearer C approaches to O, the farther does D recede from it; and when C coincides with O, D must be infinitely distant from it, or as it is often expressed, at infinity. *Pappus, VII. 160. [Corr. 2, 3 are given in De La Hire's Sectiones Conicæ, 1685, p. 3.] PROPOSITION 6. If AD, AB, AC are in harmonical progression, and the mean AB is bisected at O, then AD, OD, CD, BD are proportionals; and a result which, considering AD, CD, BD as the terms in harmonical progression, may be stated thus: The rectangle under the harmonic mean and the sum of the extremes is equal to twice the rectangle under the extremes. COR. 2. The geometric mean between two straight lines is a geometric mean between the arithmetic and the harmonic means of the same straight lines. [The arithmetic mean between two.magnitudes is half their sum.] Denote the arithmetic, geometric, and harmonic means between AD and DB by a, g, h respectively; OD. CD, .. g2 = a.h; then a = * Pappus, VII. 160. PROPOSITION 7. If AD, AB, AC are in harmonical progression, and the mean AB is bisected at O, then CB: CD = CO : CA; and conversely.* COR. 1. DB: DC = AO: AC. (De La Hire's Sectiones Conica, p. 3.) 2 AD. BC. COR. 2. AB. CD = 2 AC. BD PROPOSITION 8. AD. DB Iƒ AD, AB, AC are in harmonical progression, then AC CB = CD2; and conversely. A O C B Bisect AB at 0. Ꭰ Then AD. DB = (0D+OB)·(OD – OB) = OD2 – OB2, II.5, Cor. and AC · CB = (OB +OC)·(OB− OC) = OB2 – OC2; II. 5, Cor. .. AD. DB – AC. CB = OD2 – 2 OB2 + OC2, = = OD2 2 OD OC+OC2, App. VI. 5 (OD - OC)2 = CD3. The theorem may also be proved without bisecting AB. II. 7 The following definitions are necessary for some of the deductions : DEF. 4.-If four points A, C, B, D forming a harmonic range be joined to another point O, the straight lines OA, OC, OB, OD are said to form a harmonic pencil. OA, OC, OB, OD are called the rays of the pencil, and the pencil is denoted by 0. ACBD. *Pappus, VII. 160. |