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eter, AD AB AE are proportionals D isto DB, si to EB, that is, the segments of
the diameter he tween iis vertices and the ordinate are tomach other a the segments of the sam belween the tangen and the Same Vertices. The demonstratio is simila to that given in the secondiarti prop. IT. b. 2 COR. 2. In the econd case, hen the ordinate ram Do the oint of Oniaci passes through the Xtremity of the second diameter, the tangent passe through the ther eXtremi tyof the fame diameter For since the distancebet Neen the ordinate and the centre is equat toth hali of the secon diameter, heres ore thedistanceae tween the tangen and the centre musti equat to the sanae semidiameter E
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me. s. I from a poliat C in an hyperbola a traight line CD be ordinatelyapplied to the diameter AB, anda traight line E e drawn frona
the sanie poliat is the semidiameter hera mea proportionalbet e en the abscissas of AB, whicli are ut offooward the
Forci CE does notriouch the hyperbola, et CP ouc it thereiare, by the preceding PrO- position, AD AB, Ainare proportionals but, by the hypothesis AD, AB, AElare proportionals whicli is absurd CE, therei ore, touches the hyperbola.
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perbola meet a transverse iameter, there ein drawn romthe oin os contacto traight
line ordinately applied to the
sam diameter the rectangle Contained by the segments of the diameter interceptedietweentheordinate and the Centre, and be- tween the ordinate and the tangent, i equat to the rectangle contained by the segments be- tween the ordinate and the vertices of the diameter and therectangle Containe by the Segments et een the tangen and the centre, and belween the tan-
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gent and the ordinate, is equat tolli rectangi contained by the Segments e tween the tangentand the vertices of the diameter. But is tho tangent meet a SeCOIad diameter, thereieinidra 1 frona the poliat os contactra traight line ordinatet applied to that se Cond diameter the rectangle contained by the Segment he-
tre, and e tween the ordinate and the tangent, is quai to the Sum of the quare of the Semidiameter and of he segmen be- tween the ordinate and the Centre and the rectangi contained by the Segments et cen thetangen and the centre. and be-
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tween the tangen and the ordinate, IS equa to the sum os ille Square of the Semidiameter and of the segments belwe ei the tangen and the ContrC.
Case 1. et the traight line hicli Ouche Fig. is. the hyperbola in C, meet the transverse iameter BAO in the poliat , and et an ordinate drawn through the poliatis Contacta thema mediameter meet it in D; then the rectangle ADEis equa to the rectangle BDO, and the rect
For since AD AB AE are proportionais, the rectangi DAE is quai to the quare fAB and these equat be in taken rom the Square of AD, the re maining rectangi ADEis equa to the re maining rectangle BDO 2. an 6. 2. Elem. NeXt, fro the Same qualS, viz. the rectangi DA and the quare of AB, talae Way the common quare 3 and 5. 2. Elem. of AE and the re maining rectangle AE is equat to the re maining rectangle BEO.
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Fig. I s. Case . et the tangen and ordinate rarento the oin C, meet the secon diameter in the potnisi re then ecause the rectangle EAD is quai to the quare of AB, adstomachos these equat the quare of AD, and the rectangi ADE, illi equat o 3. 2. Elem. thesum of the quare of AB AD. ext is tothe Same quais to wit, the rectangle EADand the quare of AB, the quare of AE eaddes the rectangi AED illle equa to thesum of he squares of AB AE. PROP. XXXVI. THEOR.