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equat therelare the quares of Κ, H areequat and there re Κ, H are likewiseequat consequently EC, DTare 26. I. Elem.)equat, and the are parallel thereser CE is half 11 def. 3. the second diameter conjugateto CD.
On the contrary the fame conStruction re-
maining is Ei half the secori diameter conjugate to CD, iis extremit E is in the by Fig. aa. Perbola adjacent to that here the potnt Wis: sor CE is equat and 11 def. 3. parallel o DF, andi is parallelao DII oheresere the trian- gles KC, H ares 26. I. Elem. equat; consequently C is equat o EF, orΗC, and ΕΚ o DH and se this reason, the rectangle ΓΚ is quanto the rectangle DUC, and thep itit reis in the hyperbola, and the potant Eb A th in the angi adjacent to that containingitis hyperbola; there re the poliati is in thea I tabent hyperbola by par 2 of the preced-
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CD in the hyperbola AD O the contrary, CE is a transverse, and CD a secon diameter conjugateri CE in the adjacent hyperbola BE. Ioin D and ME, and et E meet theother asymptot CF in P then, since E is 2 Cor def. II. . parallelo CF, and that MF is bisected in D, there fore His bisectenina, and the oin E is in the adjacent hyperbola ; there fore M touches that hyperbola :and CD is equa an parallel o ME; for CE, M are qua an parallel there fore Cinis
the second diameter conjugate to the tranSVerse
diameter CE in the ill def. hyperbola BE.
Co R. 2. The same Constructio stili re main
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I froin the extremity of a Seconditiameter of an hyperbola, astraight line e rawn parallelio an tranSverse diameter, and meeting the seconditameter Onjugate to this transverse; thesquare of the parallel, is to thoreCtangi contained by the Segments of he second diameter intercepted etween the paralleland iis vertices, a the quare fili transverse, is to the Square fili second diameter Conjugate toit anxi froni the extremit of SeCond diameter a traight linebes irawn parallel to any ther
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secon diameter, and mee tingthe transverse diameter conjugatet this ther secon diameter; the quare of the parallel, is tolli sum of the squa res of half the transverse diameter and iis segmen intercepted e tween thecentre and the parallel, a thesqua re of that ther second di ameter, is to the quare of the tranSVerSe conjugate to t.
Fig. 4. In the sirs case, et therei an hyperbola, the transverse diameter of hicli is DCd, and Iet KChi the secon diameter Conjugate toDCd, and et Bie any the secon diameter, and fro iis extremit Bira a traight line parallel to the transverse CD, and meetingi ni the second diameter conjugate to CD thesquare of L is to the rectangle Lk, a thesquare os Diito the quare o Kk.
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For the potnis B, hicli are the extremities o vertices of the econ diameters, are in the adjacent hyperbolas, of hicli the transverse diameter bis conjugate to the se-Cond I. Cor. O. . Dcla therefore the quareo BL is to the rectangle Lbas the quare of D to the quare of Κὶ 28. 3. The second case is demonstrate in the very fame manne seo the 29th prop. of this book. Coll. Hence, i sto any oint Ais an hyperbola AD, a traight line Mine drawn ordinatet applied to the right diameter Kk, and Domi an Xtremit os an secon diameter C to the fame L. a traight line Laedra wn parallel o AM the quare of BL is to the square of AM, a the rectangle ΚLbis to thesum of the quares of the semidiameter Cand the segment Mietween the Centre and the ordinate. This is evident frona the proposition, and fro the 29th of this book.
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plied to the transverse diameter Dd, and Domthe eXtremit nos the second diameter to thesame Dd, a traight line BFie drawn parallelio AE the square of BF is to the quare of AE a the sum of the quares of the semidiameter CD, and the segmen CF, et eeni the Centre and BF is to the rectangle Ed. This is demonstrated fro the proposition, and
frontali 28th of this book. PROP. XLII. THEOR.
line BF e rawn parallel tostraight lines ordinatet appliedio an diametera d, and BHiedrawn parallel to the diameter CA, hicli is conjugat lo CB, and meetin the diameter din H; the semidiameter C is