CONIC SECTIONS. 



15t 



K& 74. 



lei toT>d; and IHi perpendicular to the diameters DN, 

 d n. The triangles DKL, DHO, are manifestly similar, 

 as also the triangles dkl, dHo, and the triangles I HO, 

 iHo; therefore, DK : DH : : KL : HO, and d k : d H : : 

 fcl:Ho ; (4. 6*. E.) but because Kk is parallel to Dd, 

 DYL:DH::dk:dH (2. 6*. EX therefore, KL : HO : : 

 k I : Ho ; but HO : HI : : Ho : Hi ; therefore ex aequali 

 KL : HI : : k I: H i (22. 5. E.), but HI=H i (2. Cor. 7.), 

 therefore KL = k I, now KP = k P (8.), therefore PL 

 = El, and (34-. I.E.) ED = Ed. 



Cor. 1. Straight lines which touch a parabola at 

 the extremities of an ordinate to a diameter, intersect 

 each other in that diameter ; for K/c and Dd being bi- 

 sected at P and E, the points H, P, E, lie in a straight 

 line, (Lemma to Prop. 13. Sect. II.) 



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

 a tangent at its vertex; for if not, let a tangent be drawn 

 parallel to the ordinate ; then the diameter passing 

 through the point of contact would bisect the ordinate, 

 and thus the same line would be bisected in two dif- 

 ferent points, which is absurd. 



Cor. 3. All the ordinates to the same diameter are 

 parallel to each other. 



Cor. 4. A straight line that bisects two parallel 

 chords, and terminates in the curve, is a diameter. 



Cor. 5. The ordinates to the axis are perpendicular 

 to it, and no other diameter is perpendicular to its or- 

 dinates. This is evident from 3. Cor. Prop. 2, and 

 2. Cor. Prop. 



Prop. X. 



If a tangent at any point in a parabola meet a dia- 

 meter, and from the point of contact an ordinate be 

 drawn to that diameter, the segment of the diameter 

 between the vertex and the tangent is equal to the seg- 

 ment between the vertex and the ordinate. 



Let DH, a tangent to the curve at D, meet the dia- 

 meter EP in H, and let DEd be an ordinate to that dia- 

 meter : PH is equal to PE. For draw PK, a tangent at 

 P, meeting the tangent DH in K ; and draw iKl per- 

 pendicular to the diameter PE at i, meeting a diame- 

 ter drawn through D at I : And because Hi is parallel 

 to DI, and PK to DE, we have IK: jK::DK:HK 

 : ; EP: HP (2. 6. E.), but IK= iK (2. Cor. 7.), there- 

 fore EP=HP. 



Definition. 



10. A straight line quadruple the distance between 

 the vertex of a diameter and the directrix, is called the 

 Parameter, also the Latus Rectum of that diameter. 



Prop. XL 



If an ordinate to any diameter pass through the focus, 

 the abscissa is equal to one fourth of the parameter of 

 that diameter, and the ordinate is equal to the whole 

 parameter. 



Let DEd, a straight line passing through the focus, 

 be an ordinate to the diameter PE ; the abscissa PE is 

 equal to one fourth of the parameter, and the ordinate 

 Dd is equal to the whole parameter of the -diame- 

 ter PE. Let DK, PI, be tangents at D and P ; let 

 DK meet the diameter in K ; draw PF to the focus, 

 and DL parallel to EP. The angles KPI, IPF, being 

 equal (6), and PI parallel to EF (2. Cor. 9.), the an- 

 gles PEF, PFE, are also equal (2Q. 1. E.), and PE 

 = PF = I the parameter (Def. 1 . and 10.) Again, the 

 angle KDE is equal to LDK (6), and therefore equal 



to DKE ; consequently ED is equal to EK, or to twice Parabola. 

 EP (10.) ; therefore Dd is equal to 4 EP, or to 4 PF, "** ~ s T m ~ / 

 that is, to the parameter of the diameter, (Def. 10.) 



Prop. XII. 



If any two diameters of a parabola be produced to 

 meet a tangent to the curve, the segments of the dia- 

 meters between their vertices and the tangent are to 

 one another as the squares of the segments of the tan- 

 gent intercepted between each diameter and the point 

 of contact. 



Let QH, RK, any two diameters, be produced to Fig. ?. . 

 meet PI, a tangent to the curve at P, in the points G, I; 

 then HG : KI : : PG 2 : PP. For let PN, a semiordi- 

 nate to the diameter HQ, meet KR in O ; and let PR, 

 a semiordinate to the diameter KO, meet HN in Q ; 

 from H draw parallels to NO and QR, meeting KR in 

 L and M ; thus HL is a tangent to the curve, and HM 

 a semiordinate to KR. Now KI = KR, and KL = KM, 

 (10.) Therefore, by subtraction, LI = MR = HQ, 

 but LO = HN = HG (10.); therefore, by addition, 

 IO = GQ The triangles PGN, PIO, are similar, as 

 also PGQ, PIR; therefore GN:IO, or 2 GH:IO:: 

 PG : PI, and GQ : IR, or IO : 2 I K : : PG : PI ; hence, 

 taking the rectangles of the corresponding terms, 

 2 GH . IO : 2 IO . IK : : PG 2 : PP ; therefore GH : IK : : 

 PG* : PP. 



Con. The squares of semiordinates, and of ordi- Fig. T6. 

 nates to any diameter, are to one another as then cor- 

 responding abscissae. Let HE/;, KN/c, be ordinates to 

 the diameter PN ; draw PG a tangent to the curve at 

 the vertex of the diameter, and complete the parallelo- 

 grams PEHG, PNKI ; then PG, PI, are equal to EH, 

 NK ; and GH, IK, to PE, PN, respectively ; therefore 

 HE 2 : KN 2 : : PE : PN. 



Prop. XIII. 



If an ordinate be drawn to any diameter of a para- 

 bola, the rectangle under the abscissa and the parame- 

 ter of the diameter, is equal to the square of the semi- 

 ordinate. 



Let HBh be an ordinate to the diameter PB, the pig. y<l. 

 rectangle contained by PB and the parameter of the 

 diameter is equal to the square of HB, the semiordi- 

 nate. Let DEd be that ordinate to the diameter which 

 passes through the focus. The semiordinates DE, Ed, 

 are each half of the parameter ; and the abscissa EP is 

 one- fourth of the parameter ( 1 1 . ), therefore Dd : DE : : 

 DE:PE, and Dr/.PE = DE 2 , but D<r.PE:D..:B 

 : : ( PE : PB : : ) DE 2 : HB 2 (Cor. 12.), therefore Dd . PB 

 = HB 2 . 



Scholium. 



It was on account of the equality of the square of the 

 eemiordinate to a rectangle contained by the parameter 

 of the diameter and the abscissa, that A pollonius called 

 the curve to which the property belongs a Parabola. 



Prop. XIV. 



A straight line drawn from the focus of a parabola, 

 perpendicular to a tangent, is a mean proportional be- 

 tween the straight line drawn from the focus to the 

 point of contact, and one-fourth of the parameter of 

 the axis. 



Let FB be a perpendicular from the focus upon the Fig. *!'... 



