. Mathematics, compiled from the best authors and intended to be the text-book of the course of private lectures on these sciences in the University at Cambridge [microform]. For, by Prop. , by , by hence, by sim. tri. TC : AC :: AC : DC,TA : AD :: TC : AC or CB,TA : TD :: TC : TB,AG : DE :: CH : BI. Q. E. D. Cor. Hence TA, TD, TC, TB, 1 , . .. and TG, TE, TH, TlJ are also P™portional*. For these are as AG, DE, CH, BI, by similar triangles. PROPOSITION X. If there be any tangent, and two lines drawn from thefoci to the point of contact, these two lines will make
. Mathematics, compiled from the best authors and intended to be the text-book of the course of private lectures on these sciences in the University at Cambridge [microform]. For, by Prop. , by , by hence, by sim. tri. TC : AC :: AC : DC,TA : AD :: TC : AC or CB,TA : TD :: TC : TB,AG : DE :: CH : BI. Q. E. D. Cor. Hence TA, TD, TC, TB, 1 , . .. and TG, TE, TH, TlJ are also P™portional*. For these are as AG, DE, CH, BI, by similar triangles. PROPOSITION X. If there be any tangent, and two lines drawn from thefoci to the point of contact, these two lines will make equalangles with the II. Qq 314 MATHEMATICS, That is, the Z FET=Zf Ee. For, draw the ordinate DE, and fe parallel to Prop. V. Cor. 1, C A : CD :: CF : CA + FE,and, by Prop. VIII. CA : CD :: CT : CA ;therefore, CT : CF :: CA : CA + FE ;and, by sub. and add. TF ; Tf :: FE : 2CA + FEor f E by Prop. , by sim. tri. TF : Tf :: FE } fe ;therefore, f E = fe, and consequently Ze = Zf , because FE is parallel to fe, the Ze = ZFET jtherefore, the Z FET = Z f Ee. Q. E. D. Cor. As opticians find, that the angle of incidence isequal to thf» angle of reflection, it appears from our prop-osition, that rays of light, issuing from one focus and meet-ing the curve in every point, will be reflected into linesdrawn from the other focus. So the ray f E is reflected in-to FE. And this is the reason why the points F, f, are calledfoci, or burning points. PROPOSITION XI. If a line be drawn from either focus, perpendicular to atangent to any point of the curve, the distance ot theirintersection from the centre will be equal to the semitrans-verse axis. CONIC SE
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