. The principles of projective geometry applied to the straight line and conic . Let ABGD be the quadrangle,and E, F, G its diagonal pointstriangle. Let GE meet DA andGB in E and E. Let abed be the quadrilateral,and e, f, g its diagonal the lines joining ge to da. andcb be e and e. Projective Forms Hmmonic 83 Then projecting from E,{FECB) = {FEDA).Projecting from G, {FECB) = {FEAD).Therefore (FEDA) = (FEAD).Therefore A and D are harmonicconjugates of F and E. Given a range of three points toconstruct graphically the harmonicconjugate of any one ivith respectto the other tiuo. Then
. The principles of projective geometry applied to the straight line and conic . Let ABGD be the quadrangle,and E, F, G its diagonal pointstriangle. Let GE meet DA andGB in E and E. Let abed be the quadrilateral,and e, f, g its diagonal the lines joining ge to da. andcb be e and e. Projective Forms Hmmonic 83 Then projecting from E,{FECB) = {FEDA).Projecting from G, {FECB) = {FEAD).Therefore (FEDA) = (FEAD).Therefore A and D are harmonicconjugates of F and E. Given a range of three points toconstruct graphically the harmonicconjugate of any one ivith respectto the other tiuo. Then taking sections on e, {fech) = (feda).Taking sections on g,(fecb)==(fead).Therefore (feda) = (fead).Therefore a and d are harmonicconjugates of/and e. Given a pencil of three rays toconstruct graphically the harmonicconjugate of any one luith respect tothe other Let A,B,Cbe the given is required to construct theharmonic conjugate of C withrespect to A and B. Take any two points 8 and Scollinear with G and join S and Sto A and B. Let AS. SB andSB. 8A be R and A. Then fromthe harmonic property of a com-plete quadrangle the line ABmeets the base in the requiredpoint D. Let a, b, c be the given lines. Itis required to construct the har-monic conjugate of c with respectto a and b. Take any two lines s and sintersecting on c and construct theintersections of s and s with a and as. sb and sb. sa be b anda. Then from the harmonic pro-perty of the complete quadrilateralthe line joining ab to the vertexis the required ray d. 47. To find the common pair of harmonic conjugates of tiuo given pairsof collinear points A, A and B, B. Take any point K not on the straight line ABBA. Describe circles through AKA and BKB. These circles will 84 Principles of Prqjective Geometry intersect in a second point L. Let KL meet ABBA in 0. Draw atangent OT
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
