140 



CONIC SECTIONS. 



BHlpse. hence the figures DH/jC, H Cd k are parallelograms 



1 r (S3, and 34-. 1. E.) and as the angles DC// and C // H, 



that is (29. 1. E.) the angles DC //and/; C r/ are equal to 

 two right angles, DC and C d lie in one and the same 

 Straight line (14 1 E.) therefore DCrfis a diameter, 

 and as D C=HA=C d (53 1. E.) the diameter is bi- 

 sected at C 



Prop X. 

 The tangents at the vertices of any diameter are pa- 

 rallel 



%. 13. Id Df/ he a dhrrrter, and HDK, hdk tangents 



at its vertices Draw straight lines from D and d to 

 F and / the foci. Tic triangles FCD, / C d are in all 

 respects equal, for FC=/'C, CDzsCd', and the angles 

 FCD, / C d are equal ; therefore the angles CDF, C d f 

 are eqiial, and the lines DF, d/are parallel (29. I.E.) 

 and since DF=d/] the figure FD fd is a parallelogram, 

 of which the opposite angles D, dare equal (34. 1. E-) 

 Now the angles FDH, fd //, are half the supplements 

 of these angles (8.) therefore the angles FDH, fdh 

 are equal; and hence CDH, Cdfi are also equal, con- 

 sequently HD is parallel to h d. (27 1 E.) 



Cor. 1 If tangents be drawn to an ellipse at the 

 Vertices of a diameter ; straight lines drawn from either 

 focus to the points of contact make equal angles with 

 these tangents. For the angle FDH is equal toFcJA. 



Cor. 2. The axes of an ellipse are the only diameters 

 which are perpendicular to tangents at their vertices. 

 For let D d be any other diameter, then FD, F d are 

 necessarily unequal, and therefore the angles FD d, 

 F d D are unequal, and adding the equal angles FDH, 

 F d A, the angles HD d, A /■/ D are unequal, therefore 

 neither is a right angle. (29. 1. E.) 



Prop XI. 



A straight line drawn from either focus of an ellipse 

 to the intersection of two tangents to the curve, will 

 make equal angles with straight lines drawn from the 

 same focus to the points of contact. 



Kg. 14. Let HD, H d be tangents to an ellipse at the points 



D, d, let a Straight line be drawn from H their inter- 

 section to F either of the foci, and let FD, F d be 

 drawn to the points of contact, the lines DF, d F make 

 equal angles with HP. For draw Df, df to the other 

 focus, and in FD, Fd produced, take DKrrD/'and 

 dk=zd i ; join HK, HA, and draw fK, fk meeting the 

 tangents at G and g. The triangles/DH, KDH have 

 D/'— DK : and DH common to both, also the angle 

 /DH equal to KDH (8 ) therefore f H=KH. In like 

 manner it may be shewn that f"H=fc H ; therefore 

 H ¥ = H A. ow FK = FA, for each is equal to FD + D F, 

 or F . -\-df, that is to the transverse axis (1.) there- 

 fore the riangles FKH, F AH are in all respects equal, 

 and hence the angle KF II is equal to A FH, there- 

 fore DF and d F make equal angles with HF. 



Cor. 1. Perpendiculars drawn from the intersec- 

 tion of two tangents to straight lines drawn from either 

 -focus through the points of contact are equal. For HI, 

 H /, perpendiculars to FD, F d are manifestly equal 

 (26. 1. E ) 



Cor. 2. Straight lines drawn from the intersection 

 of two tangents to the foci make equal angles with 

 the tangents. For the angles FHK, FH k being equal, 

 and FHK=FHD-f DHK=FHD-f DH f=2 FHD + 

 FH /; and in like manner FH A = FH// -f- d H f- 

 2fHd + FH /, therefore 2 FHD + FH/ = 2 i /H</-j- 

 FH/ and 2 FHD=2/H d, and FHD=/H d. 



Prop. XII. Eiiipwi 



If there be two tangents at the extremities of a chord 

 in the ellipse, and a third parallel to the chord, the 

 part of this tangent intercepted between the other two 

 is bisected at the point of contact. 



Let HD, H d be tangents at the extremities of the F 'l'« 15 * 

 chord D d, and KPA another tangent parallel to the 

 chord; and meeting the others in K and A ; the line 

 K A is bisected at P, the point of contact. From the 

 points of contact D, P, //, draw lines to F, either of the 

 "foci ; and from the intersections of each two tangents, 

 draw perpendiculars to lines drawn through the focng 

 from the points of contact, that is draw Hi, H i per- 

 pendicular to FD, F d; and KM, K •< perpendicular 

 to FD, FP; and AN, kn, perpendicular to F d, FP. 

 The triangles DHI, DKM are manifestly equiangular ; 

 as also ci H i, d k N ; therefore, DH : DK : : HI : KM 

 (4. 6. E ) and dVL -: d k: : H i : AN ; but K A being pa- 

 rallel to "Dd; DH:DK::c'H:dA; therefore HI: 

 KM:: Hi: AN; now HI=Hi (Cor. 1. 11.) therefore 

 KM— A N, but KM=K m, and k N=A n (Cor. 1. 11.) 

 therefore K mszlc n, and since we have manifestly Km: 

 A n : : KP : A P, (4. 6'. E.) therefore KP=A P. 



Lem-ua- 



Let HK'A' be a triangle, having its base K'A' bisect- fig, is. 

 ed at p ; and let K A, any straight line parallel to the 

 base, and terminated by the sides, be bisected at P, 

 then the points p, P, and the vertex of the triangle are 

 in the same straight line : and that line bisects Dd, any 

 other straight line parallel to the base. 



Complete the parallelograms IIK'/T, HKPS. The 

 triangles, HKA, HK'A' being similar, and KA, K'A' si- 

 milarly divided at P and p, we have HK : HK' : : K A: 

 K' A' : : KP : K'p; hence the parallelograms HKPS, 

 HK'wT are similar. Now they have a common angle 

 at H ; therefore they are about the same diameter, 

 (26. 6. E.) that is, the points H, P, p, are in a straight 

 line. And if Dd meet that line in E, we have KP : DE 

 (: : PH : EH) : : PA : E 0, therefore DE = Etf. 



"Definitions, 



12. Any chord not passing through the centre, 

 which is bisected by a diameter, is called an ordinate 

 to that diameter. 



13. The segments into which an ordinate divides e 

 diameter are called Abicissee. 



Prop. XIII. 



Any-chord not passing through the centre, bu' pa- 

 rallel to a tangent, is bisecte I by t T e diameter which 

 passes through the point of contact : or it is an ordi- 

 nate to that diameter. 



The chord DE d, which is parallel to K It, a tangent pig-. V7. 

 at P is bisected at E by the diameter P p. Draw K'p A' 

 a tangent at p, the other end. of the diameter ; and DH, 

 rfH tangents at D,rf, the extremities of the chord 

 meeting the other tangents in K , A, and K'A'. Then 

 KP A and K'p A' are bisected at P.andp-(12.) there- 

 fore the diameter P/),when produced, must pass through 

 H, and bisect D d, which is parallel to K' A' or K A in E 

 (Lemma.) 



Cor. 1 . Straight lines which touch an ellipse at the 

 extremities of an ordinate to any diameter intersect 

 each other in that diameter. 



Cor. 2. Every ordinate to a diameter is parallel to 

 a tangent at its vertex. For if not, let a tangent be 



