. The principles of projective geometry applied to the straight line and conic . Method of False Positions 239 Hence the ranges described bj^ P and S are projective and their self-corre-sponding points give possible vertices of the required triangle. The correlative construction gives the solution of the following problem : To circumscribe a triangle to a given conic so that its vertices shall lie on threegiven lines. The above and likewise the correlative solution may be extended to the case inwhich the triangle is replaced by a polygon, whose sides pass through fixed pointsor whose vertices
. The principles of projective geometry applied to the straight line and conic . Method of False Positions 239 Hence the ranges described bj^ P and S are projective and their self-corre-sponding points give possible vertices of the required triangle. The correlative construction gives the solution of the following problem : To circumscribe a triangle to a given conic so that its vertices shall lie on threegiven lines. The above and likewise the correlative solution may be extended to the case inwhich the triangle is replaced by a polygon, whose sides pass through fixed pointsor whose vertices lie on fixed lines. {d) To describe a circle to tov-ch three given circles. Si ^ :^^ Sa. T It has been proved (1) that the six centres of similitude of three circles lie three by three on fourstraight lines termed the axes of similitude. (Ex. 18, Chapter xi.) (2) that if a circle touch two given circles its points of contact are collinearwith a centre of similitude of the circles. (Ex. 15, Chapter xi.) Construct .Sl, .5^2, ^3 three collinear centres of similitude of the three givencircles. Take P any point on the circle centre (\. John P to So to meet circlecentre C3 at Q. Join Q to .S, to meet circle centre C-i at R. Join R to S-^io meetcircle centre C^ at T. Then (Art. 84 (11)) for ditterent positions of P the ranges 7*, R, Q, P on thecircles are projective. Determine the self-corresponding points of the ranges T andP. Let P, Q, R be the positions of P, Q, R, when F is one of these points. Since the triangles PQR and ^ are in perspective PC\., QC:^, RCU meetat a point 0. This point is the centre of the circles, which by the converse of(2) touch the given circles in
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
