. The principles of projective geometry applied to the straight line and conic . 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
. The principles of projective geometry applied to the straight line and conic . 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 Let OA, OB be the given bases, which touch the envelope at A and B. Let .S bethe point at which connectors of pairs of corresponding p»)ints subtend a constantangle. Let AS and BS meet the envelope at It and T and let the tangents at meet the bases in Q, (/ and P, P. Then the angles ASO, QS(^, PSF, OSBare all equal to some angle a. Let be 0 and be T, then by the property of inscribedquadrangle and circumscribed quadrilateral Pq, AR, OO and BT are concurrent at l\AT, QF, RB and 00 „ „ „ S. Then the angle QSR = a- RSQ = USB - BSF = USF = a- USP= USA - USP = P^.l. Therefore, since QSR= USF = PS A, the angles between three pairs of conjugaterays through S are equal and therefore (Example 13, Chapter viii) the involutionpencil at .S is orthogonal and S is a focus. CHAPTER XIV PROJECTIVE FORMS IN RELATION TO THE CONIC :—CARNOTSTHEOREM, PASCALS THEOREM, DESARGUES THEOREM 99. Carnots, Pascals and Desargues Theorems, together with theanharmonic property, are t
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