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For the extremit B of the second diameteris in the adjacent hyperbola, and Acis a l. Cor. O. . secon diameter of that adjacent hyperbolan there re H lolaches the same II des in and thereiare F, CD, CH are 33 3. proportionalS.Coll. The 35th proposition is quali true When accommodated to this Case. PROP. XLIII. THEOR.
Infrom the extremities of tW Conjugate diameter of an hyperbola, traight lines e rawn ordinately applied to any third transverse diameter the quare of the segment of the hird diameter, intercepted e tween the Centreand the ordinate rawn rom theeX tremity of the transverse diameter, is equat to the quare of tho
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ity of the ther of the Conjugate diameters together illi the Square of half the hir diameter. But the quare of the Segment of the hird diameter, interceptedietween the Centre and the ordinate rawn rom the e tremity of the se Cond diameter, is qua to the rectangle Ontaine by the segments of the fame hir diameter, et eenthe ordinat drawn rom the X- tremit of the ther of the conjugate diameters, and the Vertices of this hird diameter
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Let there e opposite hyperbolas, in hicli Fig. 24. C is the half of a transverse diameter, and et CB be the second diameter conjugate to CA, and et D be an other transverse diameter, and seo in the extremittes A, B dra AE, Fordinatet applied o Dd; the quare of EC sequa to the quare of C together illi thesquare oscin and the quare of C is quali the rectangi DE l. To the diameter dira AG parallel o BC, and H parallel o AG there fore, be-Cause of the paralleis, the triangles CAG, HBC are equi angular and since AE, F 3I. 3. are parallel, CAE, HBF re also quiangulam Consequently CE is to H F, a CA OBII, that is, a CG o CH and since Cinis a mea proportiona both between C and G, andiet ween CF and CH 33 and 42 3. CFis o CE a CG o CH and there re C is toCE a C to F consequently the quareos Dis equat to the rectangle FHci ut therectangle FH cor. 42 3. is qua to the sumo the quare of CF, CD consequently the square os Dis equa to the sum of the fame
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Squares of CF, CD talae the quare o CDfrom ach of the se quais, and there ill re- main the rectangi DE Y equa to the quare of
COR Hence, the semidiameter CD, o hichthe ordinates are drawn is to the Conjugate semidiameter Κ, a the distance bet Neen theone ordinate and the centre is to the re maining ordinate. For the square of CD is to the square of CK, a the rectangle Edris to the quare of EA, that is, accordin to the proposition, a the quare of CF is to the quare o EA; and there re C is to Κ, a CF to EA. Again, ecaus the quare o CD s. to the Square of Κ, a the sum of the quare os CF, CD to the quare of BF 41. 3. that is, by the Proposition, a the quare of C to the square
of BF the refore Cinis to Κ, as Ea BF. PROP. XLIV. THEOR.
The XC es of the quare of any conjugate Semidiameter 848 equal
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the alves of the axes, is theconjugate diameter beune quat: and i any one diameter e
cqua tociis Conjugate any ther diameter is also equato iis conjugateri and in this Case theangle Contained by the a Symptote is a right angle.
Let CA, CB e conjugate semidiameterS, Fig. 2 and CD, C the halves of the aXes, and DomA, B dra the traight lines AE, AM and BF BL ordinates to the Xes Then the eXCess of the quare o CA, CB siqua to the excess by whicli therium of the quare of CE, EA differs seo the sum of the quares of CL, LB: ut, by the preceding the quare of CEis equat to the sum of the quare of CF, CD iand by the fame propoSition the quare of Lis equat to the sum of the quares of M, CK; theres re the excessis the quare o CA, CB
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is equat to the excessi whicli the sum of the three qua res of CF, CD, E differs froin thesum of the three qua res of M, Κ, B; and the quare of CF, B are equat also the qua res of EA, M there re i theseequais e taken way the Xces by biclithe sum of the three first quare disser homilie sum of the ther three, is quai to the X-Ces by hicli the quare o CD differs stomthe quare of CK and therefore the Xces of the quare o CA, CB is quai to this ame
Fig. 25. Other ise Iet AB, AC e the alves of any two transverse diameter in an hyperbola, AD AE the asymptotes an dira the straight lines BD, CE touchiniit in theloinis B, C, and meetin the asymptotes in D, E; there- fore, by the IIth des an prop. O. of thisbook, BD is equa totalf the second diameter conjugate to AB and CE in like manner, is equa totalf the secondita meter Conjugate to AG it is tot proved that the eXCess of the square o AB, BD is equa to the Xces of the quares of AC, E.
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rallel to the asymptotes, and BG, C perpendicular to them therefore the rectan gles AFB, AH are equat l. cor. 16 3. and of Onsequence, Ficto AH, a H to FB that is, since tha triangles a re quiangular, si toFG Consequently the rectan gles AFG AH Κare equat, and thei quadruples are equat:
an since through the poliatis contacti, astraight line BF is drawn parallel to the a Symptote DF, F are equat consequently DG sequato AF, together illi FG and ence 8. 2. Elem. seu times the rectangle AF is equa to the excessis the quares of DG GA, that is, sinc the triangles DGB, AG areright-angled, to the X ces of the quare OfDB, BA It ma in the fame manne berahe Γ, that four times the rectangi AHQis equa tothe eXCes of the quares of EC, A and fourtimes the rectangi AFG, achath been proved, is equa toriour times the rectangi AHΚ; con Sequently the exces of the quare of DB,
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But i in an hyperbola an tranSverse diameter AB is qua to the secon diameter BD Conjugate to it, an other transverse diameter in the sanae hyperbola is also equa tociis conjugate secon diameter, and the angle Containe by the asymptotes is a right dingle forsince DB, A, and DF, Alare equat, and BF common, in the triangle DBF ABF theangle BFD and of consequenc the angle EAD, is a right angi : and hecause EA is right angle the angi CHE is also a rightangle and EH, H are equat, and CH conbmCn; heres ore EC, C are equat
I through the vertices of tW Conjugate diameters, seu Straight linc si drawn tota chin Conjugat hyperbolas the parallelogram formed by them is qualto that formed by the tangenis drawn through the Vertice of
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etCr S. Let AB, CD e conjugate diameterS, and Fig. 26. through thei vertice draw tangenis, me ei ingeach other in Κ, , , N and te EF, GHbe any the conjugate diameters, and throughthe vertices of these dra tangenis, me et ingeach the in , P, Q, R the figures KLMN OPQR re parallelogranas, and equa tomachother. Let S e the centre of the hyperbola ; and since both ΚΝ, Μ, and KL, Nare 3. Cor. 23. 3. parallel the figure L IN is a parallelogram. For a theaeason, PQR is a parallelogram and since ΑΚ, Κ ouch the hyperbolas in the vertices of conjugate diameters, the potnim here the meet is in an a Symptote. In like manne it ma be hewn that the est of the angies of the parallelogranas are in theasymptotes theres ore the Symptotes are thediagonals of the parallelogram S; On Sequently the parallelogram LM is the quadruple of
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the triangle SΝ, and the parallelogram OPQR the quadruple of the triangle OSR: ut thetriangles SN OS are equat, ecause therectangles KSN OS are equat 25th of thisbook, and 15. 6. Elem. in there re the paral-Ielogram ΚΙ ΜΝ, PQR re also equat. This proposition might also have been demonstrated like prop. O. b. 2. PROP. XLVI. THEOR.
I two conjugate diameter of an hyperbola me et a traight linetouching the hyperbola, the CC an e Contained by the segmento the tangen intercepted be- tween the potnti contac andine Conjugate diameterS, is equalto the quare of the semidiameter Conjugate to that diameter