. The principles of projective geometry applied to the straight line and conic . 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 Theore
. The principles of projective geometry applied to the straight line and conic . 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 in per-spective. Hence every pair of correspond-ing rays of these pencils intersects onKM. If ^i/meets the conic in U and V*,B U and B V correspond to B U and B U&nd lare self-corresponding. points of the projective ranges determinedon the conic by the pencils with verticesB and B, that is by the pairs of corre-sponding points A A, BB, CC . Hence, when the ranges are given, U and V are fixed,as is also the line KM. Similarly by considering the pencilsA. and J. ^ it is seen thatthe point L{AC. CA) is on UV. HenceK, L, M are collinear. This proof should be compared withthat given (Art. 36) for the correspondingtheorem in which the vertices of thehexagon lie on a pair of straight lines. * This proof only holds when the lineKM meets the conic in real points. required to prove that these lines areconcurrent. The two groups of three tangentsa, 6, c and a, b, c determine two pro-jective systems of tangents to the conic.(Art. 95 (a).) Consider the ranges b. abc andh. abc. They have a self-correspondingpoint bb and are therefore in every pair of corresponding
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
