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hyperbola, and is bisected in thepol ni here it meet it this line ouches the hyperbola: andis it ouch he hyperbola, it is
bisected in the oin os contact.
Let thereae an hyperbola the asymptotes of Fig. 15.whicli are AB, AC, and let in traight in n. BC, terminate by the asymptotes, meet it in theloin D, andie bisected in D; the straight Iine BC ouches the hyperbola. Through D dra D parallel to the oneasymptote AC, and meetin the ther in E; and in C ahe any oin G, through,hicli dra GH parallel o DE GH, ill meet the hyperbola l8. . in ome Ointi then, he- cause BD, DC are equat BE EA re also equat; and because of the quiangula trian- gles, Dis to ED, a BH to G there re l. 6. Elem. tho rectangle BEA is to the rectangi DEA, a the rectangle H to GHA: hut the rectangle BEA is 5. 2. Elem. greater
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than ΒΗΛ there re the rectangi DEA salso greater l4. 5. Elem. than the rectangle GHA; that is, ecausea is in the hyperbola, the rectangi FH is greater than the rectangle GH , and theres ore FH is greater than H; there re the poliat G is ithout the hyperbola; and there sore the traight line Ctouches the hyperbola in the potia D.
pist. s. Is a traight line M. termina ted ' by the asymptotes, is iis ected by the hyperbola in the oin D, it ouches the hyperbola in his
It is lain that the traight line inpasses no with in the hyperbola; for fit passed, ith- in the hyperbola it m ould necessarii meet itaga in in another potnt, ecause the potnisi, M are ithout the hyperbola: ut it is impossibi forcito meet the hyperbola in an other potnt ut D. For, i possibie, et i meet it
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likewis in N; there re M is l4. 3. equalto DL, that is, accordinito the hypothesis, toDM : hicli is absurd There re LM alis not illi in the hyperbola, nor meet it nywhere ut in the poliat D and therei ore LM touches it in D. O the contrary is the strat glit line LM, termina ted by the asymptotes, ouch the hyperbola in D, it is bisected in the oint of con
bola; and there fore, contrar to the hypothesis LM cut the hyperbola. COR. 1. Hence through the Same Potnti an Fig. 13. hyperbola, ni one stra ight line Cani drawn n. i.
to uching the hyperbola. Let oint in thes hyperbola, and through thni potn to the asymptote AB ra
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let i meet the asymptote AC in then, sinceBE, A are equat, BD, DC are also equat BC, heresere, ouches the hyperbola in D. Andio the straight line an touch it in the fame potiatici for, i possibie, et LDM alsotouch it then, since BE, Exare equat, here- fore LE, A re nequat an Consequently LD, M are likewis une quat theresere L Idoes notriouch the hyperbola. CoR. 2. Henc is manifest, he manne bywhich, i the asymptotes AB, AC of an hyperbola be ive in position a stra ight line BC an e drawn, hich hali ouch the hyperbola in a givera potnt D. Fig. l5. COR. 3. I through the vertices os a trans-n 2. verse diameter two traight lines e rawntouch in the hyperbolas, the are parallel toeach other. Le AC, Boebe the asymptotes, and et OB QPR ouch the hyperbolas in the vertices of the transverse diameter CP; the tangent AB QR are parallel. Dra to
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et ther asymptote A the straight lines S, PT parallel to the ther, and the triangle SCO, TCP are equiangular by the proposition, AO,
OB are equat, and ecause of the paralleis,AS, SC are also equat and in the manner,
to CT, an Consequently, a C to Q; there- fore the triangle OCA, C are equiangular; and there fore Α, PQ are parallel. COR. 4. Andri a traight lineae drawn parallelao a tangent, and meeting the hyperbola; the quare of the segment of the tangent be-4ween the poliat os contac and either of the asymptotes, is quai to the rectangle Contained by the segments of the parallel between ei therpointi conco urse with the hyperbola, and thea Symptotes . For his rectangle is equa to therectangle Contained by the segments of the tangent i5 3. be tween the poliati contactand the asymptotes that is equa to the Squareo iis segmentiet ween the potnt of contactand either of the asymptote S.
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Fig. s. The a Symptote AB, AC Dan hyn se perbola, and a ioin F in the Same, e in gi ven in position to dra a traight line hichshali ouch the hyperbola, and be parallel ora traight in KO, whicli salven in position, and Cut both the asymptotes of the hyperbola, o Opposite hyperbo