. The principles of projective geometry applied to the straight line and conic . the pencil formed by their polars. If Q\, Q2, Qs, Q4 are four collinear points situated on a line^, theirpolars, which all pass through the pole of p, meet p in points Q/, Qo,QI, Qi, which are conjugates of Qi, Q2, Q3, Q4. Hence the rangeQ1Q2Q3Q4 is projective with the range Q^Q^QIQI and therefore withthe pencil formed by the polars of Qj, Q2: Qs, Qi- The anharmonic ratio of four points on a circle is equal to that ofthe four tangents at these points. Let A, B, C, D be any four pointson the circle and a, b, c, d t
. The principles of projective geometry applied to the straight line and conic . the pencil formed by their polars. If Q\, Q2, Qs, Q4 are four collinear points situated on a line^, theirpolars, which all pass through the pole of p, meet p in points Q/, Qo,QI, Qi, which are conjugates of Qi, Q2, Q3, Q4. Hence the rangeQ1Q2Q3Q4 is projective with the range Q^Q^QIQI and therefore withthe pencil formed by the polars of Qj, Q2: Qs, Qi- The anharmonic ratio of four points on a circle is equal to that ofthe four tangents at these points. Let A, B, C, D be any four pointson the circle and a, b, c, d the tangentsat these points. Let P be any otherpoint on the circle and let p the tan-gent at this point meet a, b, c, d inA\ B, C, D. Then the anharmonic ratio of A,B, G, D is that of the pencil (P. ABGB)and that of the tangents a, b, c, d is{ABGD). Since A, B, G, D arethe poles of PA, PB, PG, PD, theseanharmonic ratios are equal. 78. Properties of Pole and Polar. The following is a restatement of the theorems of the last twoArticles together with certain important The Circle 135 (a) The line joining two con-jugate points, if it meets the curvein real points, is divided harmonic-ally by the curve. For the polar of a point is the locusof points which are harmonic conjugatesof that point with respect to the pointsof intersection of any chord throughthat point with the curve. (b) The envelope of the polarsof all points on a straight line isthe pole of the line. For the pole of a given line is on thepolars of all points on the given line andtherefore all the polars must contain thepole. (c) The pole of the connectorof two points is the point of inter-section of their polars. {d) The polar of a point onthe curve is the tangent at thepoint. (e) Any point on a tangentis a conjugate of the point ofcontact of the tangent. (/) The polar of a pointdetermines on the curve the pointsof contact—real or imaginary—ofthe tangents from the point to thecurve. For each point of con
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
