A first course in projective geometry . the theorem must betrue for any conic. Cor. 1. Any range of collinear points is homographic withthe pencil formed by their polars with respect to a conic. Note. The pencils C(PQRS) and pqrs are actually equal, superposable in the circle. This property, however, is notpreserved after projection in general, for equal angles donot project into equal angles. PROPERTIES OF THE CONIC 213 Cor. 2. The cross-ratio of the pencil formed by any fourdiameters of a conic is equal to that of the pencil formed bytheir four conjugates. (h) The intersections of fourfi
A first course in projective geometry . the theorem must betrue for any conic. Cor. 1. Any range of collinear points is homographic withthe pencil formed by their polars with respect to a conic. Note. The pencils C(PQRS) and pqrs are actually equal, superposable in the circle. This property, however, is notpreserved after projection in general, for equal angles donot project into equal angles. PROPERTIES OF THE CONIC 213 Cor. 2. The cross-ratio of the pencil formed by any fourdiameters of a conic is equal to that of the pencil formed bytheir four conjugates. (h) The intersections of fourfixed tangents to a conic withany fifth tangent form a rangeof constant cross-ratio. § 2. Props, (a) The joinsof four fixed points on a conicto any fifth point on the curveform a pencil of constant cross-ratio. We shall prove the theorems for the circle and deduce forthe conic by projection. Let A, B, C, D be the fixed points on the circle (Fig. 107a). Consider the pencils obtained by connecting these to anypoints V, V on the circle. V. Then Fig. 107a. sin AVB . sin CVD VJABCD} =^—--T^r,^ ^ sm AVD . sni CVB and the angles AVB, CVD, AVD, CVB are either equal or supplementary to the angles AVB, CVD, AVD, CVB. In either case the sines of these angles are equal each toeach. Hence V{ABCD} =V{ABCD}, and theorem (a) is proved forthe circle. Projecting and applying the same argument as in §1, itholds likewise for the conic. 214 PROJECTIVE GEOMETRY To prove (6), let a, b, c, d, v be tangents to the conic atA, B, C, D, V respectively, the first four being fixed in eachcase, and V and v variable (Fig. 1076).
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