A first course in projective geometry . n rVy^ sm cVa;2. sin c\/x-^ sm aV^y^ • sin aV^jan expression involving the mutual inclinations of the rays only. _,.... AZ^.AZ, BX^.BX, CY^.CY, ., Similarly ^^^ • ^^-^ • ^^77^^- = ^^^ ^^^ expression, since BVXj and h\/x^ are the same angle, etc. .*. the two products are equal. But the small-letter product= unity for the circle. .*. the required result follows for the conic. CARNOTS THEOREM 125 § 2. The theorem of the last article is very important andadmits of a number of applications. Perhaps the most usefulof these is (the so-called) Newtons theorem, w
A first course in projective geometry . n rVy^ sm cVa;2. sin c\/x-^ sm aV^y^ • sin aV^jan expression involving the mutual inclinations of the rays only. _,.... AZ^.AZ, BX^.BX, CY^.CY, ., Similarly ^^^ • ^^-^ • ^^77^^- = ^^^ ^^^ expression, since BVXj and h\/x^ are the same angle, etc. .*. the two products are equal. But the small-letter product= unity for the circle. .*. the required result follows for the conic. CARNOTS THEOREM 125 § 2. The theorem of the last article is very important andadmits of a number of applications. Perhaps the most usefulof these is (the so-called) Newtons theorem, which may beenunciated as follows: If chords be drawn through any point in the plane of aconic to meet the curve, the ratio of the rectangles of theirsegments depends only upon the directions in which thechords are drawn, not upon the position of the particularpoint through which they are drawn. ^^e have to show, in fact, that if TPP, tijp and TQQ, tqq^(Fig. 63) be any two pairs of parallel chords, then ^TQ,.TQ~ Fio. 63. Let w, %d be the infinitely distant points of intersectionof the parallels, and let qtl meet PP at \i and fp meet QQat 11. Then, applying Carnots theorem to the triangle tuw^ uq. uq t]). tp tvP. wPtq . tq urp . wp uP. uP But wP = v:^ and wP = top. tp. ip _ wP. uq . u^ 126 PROJECTIVE GEOMETRY Similarly, applying the same theorem to the triangle Juw\we have jP. TP uq. uq wQ,. wQ^ _ uP. uP wq . wq TQ. TQ Also wq^ = wQ and tvQ, = :wq\ •• TP. TP uP uq .uP TQ. TQ . iiq . TP. TP tp. ¥ TQ . TQ tq . tq Cor. If the chords move parallel to themselves, until theybecome tangents, the rectangles of the segments of the chordsbecome the squares on the parallel tangents. Hence the ratio of the rectangles of the segments of thechords = the ratio of the squares on the parallel tangents. § 3. We proceed to apply this to some very importantparticular cases. It has been shown that in a central conic a diameter bisectsall chords parallel to its
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
