. The principles of projective geometry applied to the straight line and conic . (2) To describe a conic through a given point to determine given involutions ontwo given Let A be the given point and I and I the lines on which the involutions aresituated. Let I and I meet at 0 and let 0 and 0 be the conjugates of 0 in the twogiven involutions. Then OO is the polar of 0. Join OA to meet OO in K, Take A the harmonicconjugate of A with respect to OK. Then A is a point on the P. G. 22 :338 Principles of Projective Geometry Let *Scand ,S be the two points on OO which are centres of
. The principles of projective geometry applied to the straight line and conic . (2) To describe a conic through a given point to determine given involutions ontwo given Let A be the given point and I and I the lines on which the involutions aresituated. Let I and I meet at 0 and let 0 and 0 be the conjugates of 0 in the twogiven involutions. Then OO is the polar of 0. Join OA to meet OO in K, Take A the harmonicconjugate of A with respect to OK. Then A is a point on the P. G. 22 :338 Principles of Projective Geometry Let *Scand ,S be the two points on OO which are centres of perspective for thetwo invohitions (Art. 60). Then the triangle OSS is self-conjugate with regard tothe conic (Art. 148). Join ..-l to S and S to meet the sides of this triangle in A^ and A^. Let B and C be the harmonic conjugates of A with respect to AyS and vl2S. Then B and C are on the conic. Let BA meet I in U. The polar of U is the line joining its conjugate on V to itsharmonic conjugate with regard to BA. Let this line meet UA in R. Let U bethe harmonic conjugate of A with respect to UR. Then U is also on the curve. Hence the five points A, A, B, G and U
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
