. The principles of projective geometry applied to the straight line and conic . onic 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 o
. The principles of projective geometry applied to the straight line and conic . onic 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 to one of the circles andmeasure off distances OE and OF onA A equal to OT. ThenOE- = OF- = OT^ = Hence E and F are harmonic con-jugates of A A and of BB and aretherefore the required points. It should be noticed that (1) E and F are uniquely determined; (2) E and F are real if 0 is outside the circles and, for this to bethe case, the segments A A and BB must not 48. Conjugate Points—Pole and Polar. Conjugate points with respect to twogiven straight lines are such that theirconnector is cut harmonically by thepair of lines. Thus, in the figure if {ABGD)=-\G and D are conjugates with respect toa and b. Conjugate lines with respect to twogiven points are such that they areharmonic conjugates of the lines joiningtheir point of intersection to the givenpoints. Thus, in the figure if {abed) = — 1a and h are conjugates with respect toC and D.
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