156 



CONIC SECTIONS. 



Parabola, rectrix in Q ; a line joining ^ and D shall be a tangent 

 V "~""~Y~"'"' at D ; and only one tangent can be drawn at that point. 

 It is evident from Case 3, and Corollary 3, that DQ 

 is a tangent ; and to prove that no other can be drawn 

 ?ig, G£. ^ D ; (Fig. 68.) draw DHQ' any other line, meeting 

 the directrix, and its parallel in Q' and H' ; draw QIC 

 perpendicular to FD, and take FL=FQ' ; then, be- 

 cause FH=Q'K (4.) and FL=Q'F (by construction) 

 therefore FH is less than FL ; hence the line Q'D 

 cuts the parabola, as appears from the first corol- 

 lary. 



Cor. 5. Tangents drawn at the extremities of a 

 chord passing through the fccus, meet in the directrix, 

 and form a right angle ,• and a perpendicular to the 

 chord at the focus meets the directrix, at their intersec- 

 tion. See Fig. 66. 



Prop. VI. 



A tangent to a parabola makes equal angles with a 

 straight line drawn from the point of contact to the 

 focus, and another perpendicular to the directrix. 



"Fig. 69. 



Fig. 70. 



If the tangent be at the extremity of the axis, the 

 truth of the proposition is evident from Cor. 3. Prop. 2. 

 If ithave any other position, as DQ,let itmeet the direc- 

 trix in Q, and draw DF to the focus, and DE perpen- 

 dicular to the directrix; And because FD=DE and 

 QD is common to the triangles QFD, QED, and the 

 angles at F and E are right angles (by Hyp. and 4. Cor. 

 5.) the triangles are in all respects equal (47. and 8. 1. 

 E.) hence the angles QDE, QDF are equal. 



Cor. J. The straight line FE, which joins the points 

 F and E, is perpendicular to the tangent, and is bi- 

 sected by it. For the triangles DEI, DFI, are evi- 

 dently equal in all respects (4. 1. E.) 



Cor. 2. Every tangent, except that at the vertex, 

 meets the directrix at an oblique angle. 



Definition. 



7- A straight line parallel to the axis, which termi- 

 nates at one extremity in the parabola, and lies entire- 

 ly within it, is called a diameter, and the point in which 

 :it meets the curve is called its vertex. 



Prop. VII. 



A straight line drawn from the focus of a parabola 

 to the intersection of two tangents, makes equal angles 

 with straight lines drawn from the focus to the points 

 of contact. 



Let tangents to a parabola at D and d intersect each 

 other at H ; draw DF, d F, HF, to the focus ; the lines 

 DF, d F make equal angles with HF. 



Draw DE, de perpendicular to the directrix, and join 

 HE, He. The triangles HDE, HDF are in all respects 

 equal, (4. 1 . E.) for they have DE = DF, and DH 

 common to both ; and the angles at D equal ; there- 

 fore HE=HF, and the angle HED is equal to HFD. 

 In the same way it may be proved, that the triangles 

 tide, H dF are equal, and consequently that HF=He, 

 And the angle HF d to the angle H e d ; but HE being 

 equal to H e, for each has been proved equal to HF, 

 the angles HE e, H e E are equal, (5. 1. E.) and adding 

 the right angles eED, E ed; the angles HED, H e d are 

 equal ; but these have been proved equal to HFD, HFd, 

 therefore the line HF makes equal angles with FD, F d. 



Cor. 1. Perpendiculars drown from the intersection 

 of two tangents to lines drairn from the focus through 



the points of contact, are equal. For HI, H i, be- Parabola, 

 ing drawn perpendicular to DF, dF; the triangles HFI, *"■" "Y""*"' 

 HF i are manifestly equal (26. 1. E ) and therefore 

 HI=H i. 



Cor 2. Perpendiculars from the intersection of two 

 tangents to diameters passing through the points of 

 contact are equal. 



Draw GH g through H perpendicular to DG, dg; 

 and because the triangles HDG, HDI have HD com- 

 mon to both, the angles at D equal, and the angles at 

 G and I right angles, they are in all respects equal, 

 (26. I.E.) andHGrrHI; in like manner it appears 

 that Hg=H i; but^HI = H i, therefore HGzrHg. 



Cor. 3. If a straight line HF be drawn to the fo- 

 cus, from the intersection of two tangents HD, H d, 

 and another HK be drawn perpendicular to the direc- 

 trix ; these will make equal angles with the tan- 

 gents. 



The triangles EHK, eHK are equiangular (32. 1. E.) 

 for the angles at K are right angles, and the angles 

 at E and e equal (5. 1. E.) therefore the angles EHK, 

 eHK are equal; but EHK=EHD + DHK=DHF + 

 DHK = FHK+2DHF; andeHK=eHo!-t- dHK= 

 rfHF-frfHK=FHK + 2f/HK,thereforeFHK + 2DHF 

 =FHK +2d HK, and 2 DHF =2 d HK, or DHF = 

 dHK. 



Prop. VIII. 



If two tangents be at the extremities of a chord, and 

 a third tangent be 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 Fig- 7'- 

 chord Dd, and Kpk a tangent parallel to the chord, 

 meeting the others in K and k ; the line K k is bisect- 

 ed at P, the point of contact. From the intersection of 

 each two tangents, draw perpendiculars upon the dia- 

 meters passing through their points of contact, that is, 

 draw HI, Hi perpendicular to the diameters DL, dl, 

 and KM, Km perpendicular to the diameters DL, PE, 

 and 4N, kn perpendicular to the diameters dl, PE. 



The triangles DHI, DKM are manifestly equiangu- 

 lar, as also the triangles d H i, d k N ; therefore HD : 

 DK::HI:KM, and H d: dk: : H i: k N, (4. 6. E.) 

 but because Kk is parallel to Dd, HD : DK ::Hd:d& 

 (2. 6. E.) therefore HI : KM ::Hi:k N, now HI=H i 

 (2. Cor. 7.) therefore KM=AN; but KM = Km, and 

 KN=£« (2 Cor. 7-) therefore K m-=.k n, and since we 

 have manifestly K m : k n : : KP: k P (4. 6. E.) there- 

 fore KF=k P. 



Definitions. 



8. Any chord that is bisected by a diameter is cal- 

 led an ordinate to that diameter. 



Q. The segment of a diameter between its vertex and 

 an ordinate is called an abscissa. 



Prop. IX. 



Any cliord parallel to a tangent, is bisected by the 

 diameter which passes through the point of contact, or 

 is an ordinate to that diameter. 



The chord Dd, which is parallel to the tangent KPk, Fig. ?& 

 is bisected at E by PE, the diameter that passes through 

 P, the point of contact. For, let HD, Hd, be tan- 

 gents, and DN, D» diameters at the extremities of the 

 chord, and let the tangent at P meet the former in K, k, 

 and the latter in L, /; also through H draw OHo paral- 





