. The principles of projective geometry applied to the straight line and conic . vertices are A, B, C. Since A, Bo, C^ are collinear, byMenelaus Theorem BA 1 CA CB2 ACiABo BC\ Let .1, B, C, .1, B, t be the verticesof the hexagon and let A A, BB, CCmeet the sides of the triangle ABC inA, B, C. It is required to prove thatA A, BB, CC are concurrent. Let the tangents from A, B, C maetthe sides of the triangle ABC in AiA^,B1B2, C1C2 as in the figure. Since A A, BB^, CC2 are concurrent,by Cevas Theorem ^__ 1CA ~ CBT^ AB, A Co1 BC2 * If the line LM meets the circle in real points iu the first case o
. The principles of projective geometry applied to the straight line and conic . vertices are A, B, C. Since A, Bo, C^ are collinear, byMenelaus Theorem BA 1 CA CB2 ACiABo BC\ Let .1, B, C, .1, B, t be the verticesof the hexagon and let A A, BB, CCmeet the sides of the triangle ABC inA, B, C. It is required to prove thatA A, BB, CC are concurrent. Let the tangents from A, B, C maetthe sides of the triangle ABC in AiA^,B1B2, C1C2 as in the figure. Since A A, BB^, CC2 are concurrent,by Cevas Theorem ^__ 1CA ~ CBT^ AB, A Co1 BC2 * If the line LM meets the circle in real points iu the first case or the point Ini in thesecond case is external to the conic, the construction of the corresponding figure isimaginary and this method does not give a proof of tlie theorem. 192 Similarlyand Prinrlples of Projective Geometrij Similarly ACBC AB BA., ^ 1BAi AC^-CA^BC^ ThereforeBA 4£ CBCA BC AB ACi AC^ BAi BA2 CBi CB/BCiBC^ CAi • CAo/ABi AB^ CB aW^ 1 BA, ACy CA, •BC, am] ACBC I CB,AB, BA,-CA, Therefore BA CA AC CB EC AB -1 AC, AC, BA, BA, CB, CB,BC,BC, CA, CA,AB,AB,. The expression on the right-hand sideis by Carnots theorem equal to by the converse of Menelaustheorem A, B, C are collinear. The exjjressiou on the right-hand sideis by the correlative of Carnots theoremequal to -1. Hence by the converseof Cevas theorem A A, BB\ CC proof of the converse theorems by means of Carnots theorem is left as anexercise for the student. (c) Proof by Projective ranges on the Conic. Let the inscribed hexagon be Let the circumscribed hexagon be ABCABC and let K, L, M be the ahcabc and let k, I, m be the linesIxAnts B A. A B; AC. AC find ; near and It is Pascals Theorem 193 It is required to prove that these pointsare coUinear. The two groups of three points J, B^C and A, B, C determine two pro-jective ranges on the conic. (Art. 95 (o).) Consider the i>encils andB. A BC. They have a self-correspondingray in BB and are therefore
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