A first course in projective geometry . , by considering the vanishingpoints of a pair of homographic ranges determined by a variable tangentto a hyperbola on the asymptotes. 15. If the tangent at P meet the tangents at the ends of A, A ofa diameter of a central conic in L, L, is constant. Henceprove that if any other tangent cut the tangents at A, A in M, M, thepoint of intersection of LM, LM lies on AA. 16. Prove the theorems of §2 direct from the conic by making useof § 1, Chap. XL (B. W. Home.) 17. Two pairs of tangents from a pair of conjugate points are cut byany other tangent in a
A first course in projective geometry . , by considering the vanishingpoints of a pair of homographic ranges determined by a variable tangentto a hyperbola on the asymptotes. 15. If the tangent at P meet the tangents at the ends of A, A ofa diameter of a central conic in L, L, is constant. Henceprove that if any other tangent cut the tangents at A, A in M, M, thepoint of intersection of LM, LM lies on AA. 16. Prove the theorems of §2 direct from the conic by making useof § 1, Chap. XL (B. W. Home.) 17. Two pairs of tangents from a pair of conjugate points are cut byany other tangent in a harmonic range. 18. If two conies are such that a triangle inscribed in one is circum-scribed to the other, an infinite number of such triangles exist. CHAPTER XY. PASCALS AND BRIANCHONS THEOREMS. § 1. (a) Pascals Theorem. If a hexagon be inscribed in aconic, the points of intersection ofpairs of opposite sides arecoUinear. (6) Brianchons Theorem. If a hexagon circumscribe aconic, the joins of opposite angularpoints are Fig, 115a. Let A, B, C, D, E, F be the an-gular points of the hexagon (). Let a, h, c, d, e, f be the sidesof the hexagon (Fig. 1156). PASCALS AND BRIANCHONS THEOREMS 229 Then, by the theorem of § 2,Chap. XIV., B{CDEA} = F{CDEA}. Then, by the theorem of § 2,Chap. XIV., h {cdea} =/{ cdea}(using this notation as before todenote the range determined onthe line b or/by the lines c, d, e, a).
Size: 1380px × 1810px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
