. The principles of projective geometry applied to the straight line and conic . a pair of points, the following theorems are obtained. The envelope of a line, such that The locus of a point, such that its points of intersection with a the tangents from it to a given conic given conic are harmonic conjugates are harmonic conjugates of its con- of its intersections with a pair of nectors with a pair of given points, given lines, is a conic which touches is a conic which passes through the the pair of given lines. pair of given points. 29f) Principles of Projective Geometry These theorems may be
. The principles of projective geometry applied to the straight line and conic . a pair of points, the following theorems are obtained. The envelope of a line, such that The locus of a point, such that its points of intersection with a the tangents from it to a given conic given conic are harmonic conjugates are harmonic conjugates of its con- of its intersections with a pair of nectors with a pair of given points, given lines, is a conic which touches is a conic which passes through the the pair of given lines. pair of given points. 29f) Principles of Projective Geometry These theorems may be proved independently as follows Let a and b be the given linesand p a tangent to the requiredenvelope. Then pa, pb (R and Q)are conjugate points with respectto the conic. Hence if P, the poleof p, be joined to Q and R by rand q, pqr is a self-conjugatetriangle. Let A and B be the givenpoints and P a point on the re-quired locus. Then PA, PB (qand r) are conjugate lines withrespect to the conic. Hence, ifthe polar of P meets r and g in Qand R, PQR is a Hence R is the pole of 7% andthe pencil formed by r, whichpasses through B, is projectivewith the range described by R onb the polar of B, and therefore withthe pencil q through A. Hencethe locus of P is a conic throughA and B. Hence r is the polar of R, andthe range described by R on thefixed line a is projective with thepencil described by r through Athe pole of a, and therefore with therange described by Q on b. Hencethe envelope of j) is a conic touch-ing a and b. The right-hand side of the above may also be stated as follows:The locus of the points of intersection of tangents to a given conic from pairs of conjugate points of an involution is a conic, which passes through the double jioints of the involution. Particular Cases. If the lines a and b in the left-hand figure are the connectors ofa point F to the circular points at infinity, the chord of intersection LM Theorems concerning Tivo Conies 297
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