. The principles of projective geometry applied to the straight line and conic . Draw a pair of common tangents to (1) and (2) to meet (3) in R, li and *S, ; (Art. 131) SR and SR will meet in a point U on OP and SR and *S^ ina point V on OQ. By construction U and V are conjugate points with respect to (3). But U being on POP has the same polar with respect to (1) and (3),and V being on QOQ has the same polar with respect to (2) and (3). Hence U and V are common conjugate points with respect to (1) and (2).Hence they are conjugate points with respect to all conies through A, B, C,
. The principles of projective geometry applied to the straight line and conic . Draw a pair of common tangents to (1) and (2) to meet (3) in R, li and *S, ; (Art. 131) SR and SR will meet in a point U on OP and SR and *S^ ina point V on OQ. By construction U and V are conjugate points with respect to (3). But U being on POP has the same polar with respect to (1) and (3),and V being on QOQ has the same polar with respect to (2) and (3). Hence U and V are common conjugate points with respect to (1) and (2).Hence they are conjugate points with respect to all conies through A, B, C, they are conjugate points with respect to AC and BD. Therefore thepencil {) is harmonic. (c) If a quadrangle STST he circumscribed to a conic, and on any tangent p tothe conic a point P he taken such that PS is the harmonic conjugate of p ivith regardto PT and PT, then the locus of P is a conic having double contact with the given conic. S. Theorems concerning Two Conies 289 Let p be the second tangent from P to the conic. Let p meet TS in X and letp meet TS in A. Join PT to meet 7*S in Tand TS in T. Then (P. TTXS) = {P. TTXS) by the correlative to Desargues , since the first of these pencils is harmonic, the second is also harmonic;therefore {TYXS) and {YTXS) are harmonic. Therefore since S and Tare fixed the ranges Y and X are projective,and since S and Tare fixed the ranges Y and A are projective. But since YY passes through 7, the ranges Y and Y are projective. Therefore the ranges X and A are projective and the tangents from P to theconic form two projective systems of tangents. Therefore the locus of P is a conichaving double contact with the given conic. (Art. 13L) {d) To find the locus of common conjugates of points situated on a given straightline with respect to two conies having double contact. Let the two conies (1) and (2) have doublecontact at L and M. Consider the locus ofcommon conjugates of
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