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and meetin the asymptotes in the poliat R, Q in P produced, and in ei ther directions rom the centre, take AD a me an proportional
bola in D. For since the square of AD is equalto the rectangle FR, the polia D is cor IT.
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S. in the hyperbola an since O, C are parallel, and that O is bisecte in P by thes traight line PAD, heres ore BC is bisected in D and consequently ouches the hyperbola in the sanae polia D 23. 3.)PROP. XXV. . THEOR.
I two traight lines ouch an hyperbola, O opposite hyperbolas,
and cut the asymptotes the CCtangi contained by the abscissas of the asymptotes et e en the Centre and the ne traight line, is quai to the reClangle Contain-ed by the abscissas et cen the Centre and the other straight line.
Fig. l6. et there e an hyperbola, illi AB, AD se iis asymptotes, let a traight linei touchi in C, and let another stra ight Iine GKtouchthe Same, o the opposite hyperbola. in F; the rectangles BAD, Λ willi equat.
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Fro the poliat C, F dra CH, Κ, and FL, M paralle to the asymptotes then e cause BCD ouches the hyperbola, BC is equalto CD 23. I. in and consequently B is thedoubie of AH, and A the suble of C; therei ore the rectangi BAD is the quadrupleo the rectangi CHA. I may be hewn in the Same manner, that the rectangle EAG is the quadruple of the rectangle FMAci ut Isi. 3. the rectangle H is equa to the rectangle FMA the rectangi BAD is therelare qualto the rectangle EAG. PROP. XXVI. THEOR.
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Fig. 16. et therei an hyperbola, illi AB, AD for iis asymptotes; et BD ouch it in C, and EG ouch the fame, o the opposite hyperbola, in F; oin BE, G, and F; the traight lines BE, G, and Dare parallel. Since the rectan gles BAD, AG are equat, B is to A, as G to D the refore BE, GD are parallel: oin DF, and let i meet BE in N; the since DF is to FN a GF to FE, that is, as 23 3. DC o B; therei ore ΒΝ,
CF are parallel. Co R. Heiace, o tW strat gli lines ouchinga hyperbola, thei segments bet ween the B SymplCte S, are Ut proportionalty in thelointo here the wo strat glit lines intersect achother an also in C, F the poliat os contact.
Every traight line drawn throughthe Centrem an hyperbola, and Passing ith in the angi forme l
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by the asymptotes, adjacent totha containing the hyperbola, is right diameter.
Let there e an hyperbola, o which AC, Fig. 15. BC are iis asymptotes, and dra an Stra ight -
line C through the centre, and with in theangle ACD, adjacent to the angi ACB thenis Ela right diameter. In BC produce tahe any oin D, and through D to C dra a strat glit line DF parallel to the asymptote and haviri made
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straight line drawn through the centre ofan hyperbola, bisected in the centre, an parallel o a stra ight line hichoouches the hyperbola, and equata iis segmentietween thea Symptotes, is calle the seconditameter of the diameter rawn through the oint of Con
COR. 1 mence ver se con diameter is a right diameter for it passes illi in the an gleforme by the asymptotes, adjacent to that containing the hyperbola 3. Or. I 8. 3. Co R. 2. Hence, the traight lines hich oin the vertices of a transverse diameter, and of iis Se con diameter, area parallel to the asymp
Fig. is . For let Cribe a transverse diameter, and n. a CN iis second, and AO a traight inetouching the hyperbola in the vertex of the transverse CP the traight lines Ο ΝΟ
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are 33. I. Elem an II def. . paralleloo CB, A, DEF. XII.
third proportional t two diameterS, ne of hicli is a transverse diameter, and the therit secon diameter is called the latus rectum, orthe parameter of that diameter hicli is thesrs of the three proportionals.
I from a sint in an hyperbola to tranSVerse diameter, a Stra ight line e drawn parallelo it se-Cond diameter the square of the transverse is to the quare os iisse coni diameter, a the re Clangle containe by the segments of the transversetet e en it Vertices and the parallel, is to the Square of the parallel.
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Tig. r. I et Aa be a tranSverse diameter, b iis se- con diameter, and G CF the asymptotes; froni a polia D in the hyperbola to the transverse Aa dra D parallel o Bν the quareo Aa is to the quare of Bb, a the rectangle AEa is to the quare of DE. Let D meet the asymptotes in F, G, and drais HAK ouching the hyperbola in the verteX A there re, by def. II. of this book HAis qua an paralleloo BC, and of Consequence, paralleloo FE and hecause of the equiangula triangles, the quare of CE is tothe quare o EF, a the quare of A to thesquare of AH, that is, a the fame quare ofCA to the 4 cor. 23. 3. rectangle FDG thesquare of A is, there res I9 5. Elem. to thesqua re of AH, a the rectangi AE to thesquare of ED and there re the quare of Auis to the quare of Bb, as the rectangle AEa tothe quare of ED. COR. I. The qua res of traight line drawnfro in potnts of an hyperbola, or of the opposite
hyperbola, to a transverse diameter, and pa-
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rallelo it second are to ne nother a therectan gles containe by the segments of the transverse, interceptedietween iis vertices and thos paralleis , as wayshewn in the ellipsis I. COr. 15. 2.)COR. 2. Andis the contrary is an hyper Fig. 17. bola AM, havin Aa for a transverse diame u i
ter, and Bb for iis secon diameter an is fio a potnim to the transverse Aa a straight line Et dra n parallel to the second, and meeting the transverse produce in D; anxifthe quare of A be to the square of CB, a therectangi AE to the quare of ED theloint is in the hyperbola. For since Dis parallel o BC, and consequently to HK, hichtouches the hyperbola in the verte of the transverse diameter DE, ill necessarii meet the asymptotes 3. Cor. 18. 3. and of consequenc the hyperbola, ecause the oin E is in Aa produced is, then, it oes no meet
the hyperbola in D, et it, is possibie meet ii
in anotherio in t d, o the sanae fide of Aa witirili potnt D there re the rectangle KEa suo the quare of dE, a the quare of A to the
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square o CB, that is, b hypothesis, a therectangle AE to the quare of DE; there redE, D are equat whicli is absurd There- forem meeis notulte hyperbola in , Or, asis evidenti in an potnt but D. Coll. 3. Substitute the wor hyperbola in place of ellipsis, and the third corollar of prop. I 5. b. 2. ecomes also a corollar hom his proposition
I froni a potnt of an hyperbola to
am econ diameter, a strat clinebe drawn parallelao it transverse diameter the quare of the Se- con diameter is to the quareo it transverse, a the Sum Athe quares of half the seconddiameter, and the segmen be- tween the centre and the parallel, is to the quare of the parallel.