. The principles of projective geometry applied to the straight line and conic . lly. For let CD meet {X) in A and (GDAB) is harmonic and since CD is a diameter of (Y) A and Bare inverse points with respect to ( F). Special Construction of Pole and Polar for a Circle. Let 0 be the centre of thecircle and >S the point whosepolar is required. Join OSto meet the circle at C andG and draw two chords SAAand SBB equally inclined to *SY>. Then by s^Tnmetry AB andBA will intersect at some pointS on SO and AB ami AB willbe perpendicular to SO. Hence s the polar of S will be the line throug
. The principles of projective geometry applied to the straight line and conic . lly. For let CD meet {X) in A and (GDAB) is harmonic and since CD is a diameter of (Y) A and Bare inverse points with respect to ( F). Special Construction of Pole and Polar for a Circle. Let 0 be the centre of thecircle and >S the point whosepolar is required. Join OSto meet the circle at C andG and draw two chords SAAand SBB equally inclined to *SY>. Then by s^Tnmetry AB andBA will intersect at some pointS on SO and AB ami AB willbe perpendicular to SO. Hence s the polar of S will be the line through S perpendicular toSO, Art. 76. Also because (SSCC) is harmonic and 0 is the middlepoint of CG, = OG\ while the points S, S, being conjugate points colli near with the centre,are inverse points. Hence the polar of a given point with respect to a circle is the linethrough its inveise point perpendicular to the connector of the given pointto the centre. Also: The pole of a, given line is the inverse point of the foot of theperpendicular from the centre on the given 144 Principles of Projective Geometry If AA\ BB he two pairs of collinear inverse points, ivith respect toa circle, and P be (my point on the circle, then (1) the ratio AP is constant for different positions of P; (2) the angles APB and APB are equal or supplemental. ( 53.) Let the line BAAB meetthe circle at G and D. TakeP any point on the the ranges CD A A andGDBB are harmonic. There-fore the pencils {P. CD A A)and {P. GDBB) are GP, PD are at rightangles, and are thereforecommon bisectors of theangles APA and BPE. Hence (1) AP: AP = AD : AD = a constant, (2) the angles APB and APB are equal or supplemental. 83. Coaxal Circles. If A A, BB, GG, ... are pairs of conjugate points of am. involution, thecircles desc7ihed on A A, BB, GG, ... as diameters are said to form system of circles. There are two cases to be considered according as the involutionhas, or has
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