. The principles of projective geometry applied to the straight line and conic . d K, thenOP .OP = = -OR. OK. (6) If the line p meet the circlein imaginary points, and the per-pendicular from C, the centre of thecircle, meet the line in 0, then 0 isthe centre of the involution andOP. OP = a constant. Let the polar of P meet 00 in Mand CP in N; then since the trianglesOPM and OOP are similar OM __qp OP OC .-. = But OR. 0K= OV^, where OF is the tangent from 0 to the circle. Hence if a circle be described with centre 0 and radius 0 F to meet00 at S, this circle cuts th
. The principles of projective geometry applied to the straight line and conic . d K, thenOP .OP = = -OR. OK. (6) If the line p meet the circlein imaginary points, and the per-pendicular from C, the centre of thecircle, meet the line in 0, then 0 isthe centre of the involution andOP. OP = a constant. Let the polar of P meet 00 in Mand CP in N; then since the trianglesOPM and OOP are similar OM __qp OP OC .-. = But OR. 0K= OV^, where OF is the tangent from 0 to the circle. Hence if a circle be described with centre 0 and radius 0 F to meet00 at S, this circle cuts the given circle orthogonally, and the pairsof conjugate points PP, QQ of the involution subtend right anglesat 8. S is one of the pairs of common harmonic conjugates of MC andof RK. 87. Common Involution chords of a pair of circles. Every pair of circles determines the same involution (a) on theirradical axis and (6) on the line at infinity. {a) If the circles intersect in real points L and M, the involutionon the radical axis LM in the case of each circle consists of pairs of. 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
Size: 1655px × 1510px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
