COR. 7. If a variable line meet two fixed circles in a harmonic row of points, it intersects all positions of itself homographically. [For the rectangle under its distances from the centres of the circles is constant, Art. 78, Ex. 12; therefore, etc., Cor. 4.] COR. 8. A variable line meeting two fixed circles such that the chords intercepted by them are in a fixed ratio cuts all positions of itself homographically. [By Art. 90, Ex. 8.] 143. If the distances of any point 0 from four points A, B, C, D on a line L passing through it be denoted by a, B, y, x, and the distances of any point O' measured along another line L to A', B, C, D' be similarly a', ß', y', x', the two systems of points are homographic if (B—y)(a−x)_(B′ — y')(a ́ — x ́) ̧ = which when multiplied out is of the form Axx'+Bx+Cx' +D=0,.......... .(1) an equation which enables us to determine the position of any point of either system corresponding to a given one in the other. (See Art. 140, Cor. 3.) We have seen that the lines joining corresponding points envelopes a conic touching L and L'. In the particular case when x=∞ in (1) the simultaneous value of x' is also, and the corresponding conic is therefore touched by the line at infinity. It follows obviously that when A=0 in the above equation the conic is a parabola. Thus if a variable line be drawn cutting the sides a and b of a triangle ABC in X and Y such that LAY+mBX = const., it envelopes a parabola to which the two sides of the triangle are tangents. If the axes L and L'are coincident and B=C in (1), x and a' are interchangeable in the equation and, as will be more fully explained in the next chapter, the two systems are in Involution. The double points of two systems on a common axis are found from (1) by putting x=x', in which case the equation reduces to the form Ax2+(B+C)x+D=0. EXAMPLES. 1. If the distances of two pairs of collinear points A, B and A', B' from an origin O on the line be denoted by the roots of the equations ax2+2bx+c=0 and a'x2+2b'x+c'=0, they form a harmonic row if ac'+a'c-2bb'=0. 2. Having given two of the anharmonic ratios of four collinear points equal, prove that (B− y)2(a−8)2+(y − a)2( B − 8)2 + (a− ẞ)2(y — d)2 = 0. CHAPTER XIII. INVOLUTION. 144. When of two systems of points A, B, C, ...; A', B', C', ... on any line or circle any three pairs A, A′; B, B'; C, C' which correspond are connected by a relation of the form [BCAA'] = [B'C'A'A], it has been proved in Art. 133, Ex. 9, 1°. that every four and their four opposites are equianharmonic; 2°. that A A', BB', CC', ... have a common segment of harmonic section. By Art. 140, Def., we may therefore regard either of these properties as a criterion of points in Involution. Now since [BCA'B']=[B'C'AB], by expanding and reducing we get a result previously arrived at in Art. 64, where it was shown by the application of Ceva's Theorem that a straight line drawn across a quadrilateral is cut in involution; the conjugate points A, A', etc., being the intersections of the line with the pairs of opposite connectors of the figure. Again, if a pencil of six rays be taken and a circle described through the vertex cutting the rays in points A, A'; B, B'; C, C', they form a system in involution if sin BOA' sin COB' sin AOC' 1.. ....(2) The criteria (1) and (2) are called Equations of Involution. 145. It has been noticed in Art. 134, Ex. 10, that when any two conjugates A and A' of two homographic systems are interchangeable, every two are interchangeable, and AA', BB', CC'... have a common segment or angle of harmonic section. It follows that "when any one point on an axis, or ray through a vertex, has the same correspondent to whichever system it be regarded as belonging, then every point on the axis or ray through the vertex possesses the same property." In illustration of this theorem, let the correspondents be joined in pairs to any point (A′′) on the directive axis of the systems (Art. 137). Then the corresponding rays A"B, A" B' are interchangeable, their productions through A" being A"C', A"C; therefore The locus of a point at which two homographic rows subtend a pencil in involution is their directive axis; and similarly, or by reciprocation, a variable line meeting two homographic pencils at a system of points in involution passes through their directive centre. 146. A system of points in involution on a line is completely determined when two pairs of its conjugates A, A'; B, B' are given; and the conjugate C' of any point * Townsend, Modern Geometry, vol. ii. p. 276. C is its inverse with respect to the circle described with AB and A'B' as a pair of inverse segments. If the radius of the circle is indefinitely great, one of the double points (N) is at infinity, and therefore (Art. 72, Cor. 3) MAMA', MB = MB', etc., etc.; that is, if one of the double points of a system in involution is at infinity, the segments AA', BB', CC'... have a common centre, viz., the other double point. Also a variable segment AA' of constant length moving along a given axis determines two systems of points in involution the double points of which are imaginary. 147. Theorem.-If two chords AA', BB' of a circle meet in C, any line through C which meets the circle in O and O' determines a system of points A, A'; B, B'; O, O' in involution. Let AB and 00' meet in Z (Art. 64, Iv. fig.). Then the pencil B. AB'OO' is equianharmonic with the row of points ZCOO' it determines on the transversal to it through C. For a similar reason [ZCOO']=A. BA'00′ =[A' BO’O], from which relation it follows that every four of the six cyclic points and their four opposites are equianharmonic. The concurrency of the chords AA', BB', 00', being involved in this relation, furnishes a geometrical explanation of the theorem of Art. 133, Ex. 9 (1). The following generalized statement is a direct inference of the preceding: If through any point P, inside or outside a circle (or conic) a number of chords be drawn to cut the curve in A, A'; B, B'; C, C', ..., the two systems ABC..., A'B'C'... |