. 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. VIII. TC : AC :: AC : DC, therefore, by div. TA : AD :: TC : AC or CB, and, by comp. TA : TD :: TC : TB, and hence, by sim. tri. AG : DE :: CH : BI. Q. E. D. Cor. Hence TA, TD, TC, TB, 1 are also propor-and TG, TE, TH, TI, J tionals. For these are as AG, DE, CH, BI, by similar triangles. 294 MATHEMATICS. PROPOSITION X. If there be any tangent, and two lines drawn from thefoci to the point of contact ; these
. 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. VIII. TC : AC :: AC : DC, therefore, by div. TA : AD :: TC : AC or CB, and, by comp. TA : TD :: TC : TB, and hence, by sim. tri. AG : DE :: CH : BI. Q. E. D. Cor. Hence TA, TD, TC, TB, 1 are also propor-and TG, TE, TH, TI, J tionals. For these are as AG, DE, CH, BI, by similar triangles. 294 MATHEMATICS. PROPOSITION X. If there be any tangent, and two lines drawn from thefoci to the point of contact ; these two lints will make e-qual angles with the tangent. That is, the Z TEF = For, draw the ordinate DE, andjfe parallel to Prop. V. Cor. 1, CA : CD :: CF : CA—FE,and by Prop. VIII. CAtherefore, CT : CF ::and by add. and sub. TFor/E, by Prop. VI. But, by sim. tri. TF : T/ :: FE : fe ;therefore, fE ~fc, and consequently , because FE is parallel to^-, the Ze = ZFET ;therefore, the Z FET = ZfEc. Q. E. D. CD :: CT r CA ;CA : CA —FE ;: T/ :: FE : 2CA—FE, Cor. As opticians find, that die angle of incidence is e-qual to the angle of reflection, it appears from our Proposi-tion, that rays of light issuing from one focus, and meetingthe curve in every point, will be reflected into lines drawnfrom the other focus. So the ray fE is reflected into this is the reason why the points F,/, are called fon,or burning points^ CONIC SECTIONS. 295 PROPOSITION XI. If a line be drawn from either focus perpendicular toa tangent to any point of the curve, the distance of theirintersection from the centre will be equal to the semitrans-verse axis. That is
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