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For instance, 26 THE PRINCIPLE OF DUALITY if lines a and b are coplanar, the dual of ab is a · b. However, for the sake of brevity we shall assume, until the end of Chapter 9, that all the points and lines considered are in one plane. (For a glimpse of the analogous three-dimensional developments, see Reference 8, p. ) EXERCISES 1. 12 (Reference 8, pp. 233, 446). 2. 4A. Define the further intersections A1 = BC· QR, B1 = CA · RP, A2 = BC·PS, B2 = CA · QS, = C2 = C1 AB·PQ, AB · RS. Then the triads of points A1B2 C2 , A2B1 C2 , A2B2 C1 , A1B1 C1 lie on lines, say p, q, r, s, forming a quadrilateral pqrs whose three diagonal lines are the sides b = CA, c = AB a= BC, of the triangle ABC.

Moreover, any two distinct points A and B are the invariant 48 ONE•DIMENSIONAL PROJECTIVITIES points of a unique hyperbolic involution, which is simply the correspondence between harmonic conjugates with respect to A and B. This is naturally denoted by (AA)(BB). The harmonic conjugate of C with respect to any two distinct points A and B may now be redefined as the mate of C in the involution (AA)(BB). 41 Any point is its own harmonic conjugate with respect to itself and any other point. EXERCISES 1.

5. If ABCD 7\ ABDE and H(CE, DD'), then H(AB, DD'). (S. ) 6. Let X be a variable point collinear with three distinct points A, B, C, and let Yand X' be defined by H(AB, XY), H(BC, YX'). Then the projectivity X 7\ X' is parabolic. [Hint: When X is invariant, so that X' coincides with it, Y too must coincide with it; for otherwise both A and C would be the harmonic conjugate of B with respect to X and Y. ] 7. H H(BC,AD) and H(CA,BE) and H(AB,CF),then (AD)(BE)(CF). 41 to the involution BCAD 7\ ACBE.