. The principles of projective geometry applied to the straight line and conic . of an involution, token the vertex is situated (1) on the conic or(2) on the polar of AA. BB loith respect to the conic. EXAMPLES. (1) Given six points, explain fully how with ruler and compass to construct thecentre and axis of perspective so that the perspectives of the six points shall be oneat the centre and the other five on the circumference of a circle. (2) Tangents at the ends of a focal chord of a parabola intersect at rightangles on the directrix. (3) The circumcircle of a triangle circumscribed to a par
. The principles of projective geometry applied to the straight line and conic . of an involution, token the vertex is situated (1) on the conic or(2) on the polar of AA. BB loith respect to the conic. EXAMPLES. (1) Given six points, explain fully how with ruler and compass to construct thecentre and axis of perspective so that the perspectives of the six points shall be oneat the centre and the other five on the circumference of a circle. (2) Tangents at the ends of a focal chord of a parabola intersect at rightangles on the directrix. (3) The circumcircle of a triangle circumscribed to a parabola passes throughthe focus. (4) Prove that if any point P on a conic be joined to two points B and C onthe conic by lines, which meet the tangents at B and C in C, B, and CB meets CBin T, then TP is the tangent at P. (Art. 92.) (5) If the tangents at any two points A, B on a conic meet the tangent at anypoint P \w A and B, and the lines joining A wcAtB to any point Q, on the curvemeet the directrix corresponding to a focus ;S in A, B, then the angles ASB andASB are In the fia;iire ThereforeBut and Therefore The Conic ASP=\.ASP\BSP= ASB = is the external bisector of QSASB „ „ „ „ QSB. ASB = \.ASB = ASB. 185 (6) Prove from Example 5 that the anharmonic ratio of four points on a conic isequal to that of the four tangents at the points. (7) If CA and CB be the semi-transverse and conjugate axes of a hyperbola andpoints ^and Fhe taken on the transverse axis at distances \CA^-CB- from thecentre, prove that the pairs of tangents from these points to the curve are at rightangles. (8) A point at which pairs of corresponding points of two projective ranges onfixed bases subtend a constant angle is a focus of the ccniic enveloped by theconnectors of pairs of corresponding points of the two ranges.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
