. The principles of projective geometry applied to the straight line and conic . 152 Principles of Projective (Teontetru points which are harmonic conjugates of L and M, and therefore bothcircles determine the same invohition on the radical axis. If the circles do not intersect in real points, let P be any point ontheir radical axis. The tangents from P to the two circles are equal anda circle with centre P and radius equal to these tangents intersects boththe circles at right angles. Let TT, NN be the chords of intersectionof this orthogonal circle with the two given circles. Since the radica
. The principles of projective geometry applied to the straight line and conic . 152 Principles of Projective (Teontetru points which are harmonic conjugates of L and M, and therefore bothcircles determine the same invohition on the radical axis. If the circles do not intersect in real points, let P be any point ontheir radical axis. The tangents from P to the two circles are equal anda circle with centre P and radius equal to these tangents intersects boththe circles at right angles. Let TT, NN be the chords of intersectionof this orthogonal circle with the two given circles. Since the radical. axes of the three circles taken in pairs are concurrent, the chords TT,NN intersect at some point P on the radical axis of the two givencircles. But TT, NN are the polars of P with respect to the givencircles. Hence P and P are conjugate points with respect to bothcircles and, since P may be any point on the radical axis, both circlesdetermine the same involution on their radical axis. (6) Let G and C be the centres ofthe two circles. Then G and G arethe poles of the line at infinity withrespect to these circles and pairs ofconjugate lines through G and G meetthe line at infinity in pairs of conjugatepoints of the involutions determined onthat line by the two circles. Draw any two lines at right anglesthrough G, viz. GA and GA. These arerespect to the first circle. Draw parallel lines through G. These are conjugate lines withrespect to the second circle. But parallel lines meet the line at infinityin the same points. Hence GA, GA and GA, GA determine the samepair of conjugate points o
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
