Prop. 4. If from any point of a para bola, D, (fig. 5) a perpendicular, D 11, be drawn to a diameter B H, so as to be an ordinate to it; then shall the square of the perpendicular, D H', he equal to the rectangle contained under the absciss H F, and the parameter of the axis, or to four times the rectangle II F B.
When the diameter is the axis ; let DH be perpendicular B C, join D C, and draw D A perpendicular A B, and let F be the vertex of the axis. Then, because HB= D A =D C, it follows that H D C' = DIP H Likewise, be cause B P = F C, II IP = 4 times the rectangle H FC-f-H C' (by 8. 2). Wherefore D H' H C' = 4 times the rectangle H F B Il C. ; and D H' = 4 times the rectangle H F B; that is, D = the rectangle contained under the absciss H F, and the parameter of the axis.
2. When the diameter is not the axis : let E N (fig. 3 and 4) be drawn perpendi cular to the diameter A D, and E L an or dinate to it ; and let D be the vertex of the diameter.
Then shall E N. = to the rectangle contained under the absciss, L D, and the parameter of the axis. For let D K be drawn parallel to L E, and consequently a tangent to the parabola in the point D ; and let it meet the axis in K : let E F be perpendicular A B the directrix ; and on the centre E, with the radius E F, de scribe a circle, which will touch the di rectrix in F, and pass through the focus C : then join A C, which will meet the circle again in and the right lines D K, L B, in the points P G ; and, finally, let L E meet the axis in 0.
Now since the angles C P K, C B A are right, and the angle B C 1' common, the triangles C B A, C P K are equiangular ; and A C:CB (or C K C P) : : 0 K : GP;andACxGP=OKXCH. Again, because C A= 2 C P, and C II = 2 C G, AH=2GP; and consequent ly the rectangle C A II -= CA x 2G P =OK x 2CB. But,EN'-=FA':= rectangle C A II; and consequently, =OK x 2CB = the rectangle contained under the absciss, L D, and the parameter of the axis. Q. E. D.
Hence, 1. The squares of the perpendi culars, drawn from any points of the para bola to any diameters, are to one another as the abscissa intercepted between the vertices of the diameters and the ordi nates applied to them from the same points.
2. The squares of the ordinates, ap plied to the same diameter, are to each other as the abscissa between each of them and the vertex of the diameter. For let E L, Q R be ordinates to the same diameter D N; and let E N, Q S be per pendiculars to it. Then, on account of the equiangular triangles E L N, Q R S, E L' : Q : : E N. : Q : that is, as the absciss D L to the absciss D R.
Prop. 5. If from any point of a para bola E (fig. 3 and 4), an ordinate, E L, be applied to the diameter A D; then shall the square of E L be equal to the rectan gle contained under the absciss D L, and the latus rectum or parameter of that di ameter.
For since Q R = D K, Q R' will be equal to I) M' AI K. ; but (by case 1. of Prop. 4), D M." = 4 times the rectan gle 111 Q B ; and because AI Q = Q K, M K' = 4 Al : wherefore Q = 4 times the rectangle M Q B +4M Q' ; that is, to 4 times the rectangle Q M B. But M Q =Q K=D and Al I) A ; wherefore Q R' = 4 times the rect angle It D A: and because Q E L are ordinates to the diameter A D, Q ID (by car. 2, of Prop. 4), : (: It D : L D) : : 4 times the rectangle it D A: 4 times the rectangle L D A. Therefore B L' = 4 times the rectangle L D A, or the rect angle contained under the absciss L D, and the parameter of the diameter A I) : and from this property Apollonius called the curve a parabola. Q. E. 1).
Prop. 6. If from any point of a para bola, A. (fig. 6) there be drawn an ordi nate, A C, to the diameter B C ; and a tangent to the parabola in A, meeting the diameter in D : then shall the segment of the diameter, C D, intercepted between the ordinate and the tangent, be bisected in the vertex of the diameter B. For let B E be drawn parallel to A D, it will be an ordinate to the diameter A E ; and the absciss B C will be equal to the absciss A E, or B D. Q. E. D.
Hence, if A C be an ordinate to B C, and A I) be drawn so as to make B D =_ D C, then is AD a tangent to the parabo la. Also the segment of the tangent, A ll, intercepted between the diameter and point of contact, is bisected by a tangent B G, passing through the vertex of D C.