장음표시 사용
31쪽
Is frona a poliat in a parabola a Straight line e drawn OuChing th Para bola, and is frona the fame potnia perpendicula be drawn to theaXiM; the segment of the Xis intercepted etWeen the perpendicular and the line tota chin theparabola, is bisected in the vertex of the XiS.
Let Die a poliati a parabola, and leti Fig. 5.drawn Domi ouch the parabola, and DHae Perpendicular to the Xis the segmen EH of the axis is bisected in F, the vertex of the aXis.
32쪽
Through D et D b drawn perpendiculario the directrix et DC be dra n to the focus, and et the axis meet the directrix in B and be cause the angi CD is equa to the angle ADE 9. 1. that is to the alternate angle CED, CE is equa to CD, o DA, that is, o HB and Cris equa to FB there re the re- mainde FE is equa to the remainde FH. PROP. XI. THEOR.
Every stra ight line paralleloo a straight line that ouches the parabola, and terminated both ways
by the parabola, is bisected by the diameter passing through thepoint of contaci, that is, it is ordinat ely applied to this diame
Fig. The traight lineae, hicli is terminated in n. l. 2. the pol nisi e being parallel o DK, a straight line touching the parabolas and AD, the dia-
33쪽
meter hicli passes through the potnt of contacti, meetingae in L Dis equat o Le. Let AD meet the directrix AB in fromthe pol nisi e to the directriX, draw the perpendicular EF, ef , and fro mulie focus ira C mee tingi in and stomuli centre E, distance C describet circle meet in CAagain in H this circle ill ouch the directriX in F jo in C then, ecause Acis equalto DC, and the angi AD equa to 9. 1.)CDK, D is 4. I. Elem. perpendicular o AC and the refore, eto is a right anglesto the fame AC and ecause Wis the centre of the circle FH, CG is qua to 3 3. Elem. GH jo in C and H, and C ill e equal 4. I. Elem. to H a circle, heresere, described frona the centre, and at the distance 'C, passe through H; and Maelia equa tore, it passes likewise through f theres ore, since the straight lineas ouches the circles, and the traight line AH cut them, the quare of AF is equa to the 36 3. Elem. rectan gleCΛH, that is to the quare os in there re
34쪽
ralleIs there rei is 2 6. Elem. equa to is COR. I. O the ther and is a stra ight lineae, terminate both way by a parabola, b bisected by the diameter L it is parallelto the tangent hicli passes through D, thevertex of AL is the straight line ouching the parabola in the oin D, e no parallelao LE, et another traight in be drawn ouChing the parabola, and parallel o LE the thediameter hichiasse through the poliat herethis other strat glit in tota ches the parabola, bisecis the traight line Ee: ut accordinito the hypothesis, the fame Ee is bisected by thodiameter AL hicli is absurd. Cop. 2. Ad stra ight lines ordinatet appliedio an diameter, are parallel o ne nother. Co . . I two o more paralleis e terminated both way by a parabola, the diameter whichii secis the ne or one of them, bi Secisalso the ther, o the res of them so theone that is bisected by a diameter, is parallel
35쪽
to the strat glit line ouch in the parabola in thevertex of that diameter and consequently the Other, O the thers, is, or are, parallel to the fame Straight line that ouches the parabola in that verte ; and there re is, or are, bisected by the sam diameter. COR A. An straight line, o the contrary, which bisect two parallel terminate both WaySi a parabola, is a diameter forcis it is not, it is possibi for ome ther traight linebisecting one of the parallel tot a diameter; an die ingra diameter, this other traight linemus also bisec the ther of them: ut ac- cord in to the hypothesis, the forme of thestraight lines bisecis both the parallel : hichis absurd Andri frona the potnt of contactinstraight line heirawn bisecting another straight line parallel to the tangent, an terminated both ways by the parabola, that straight line is a diameter forcis it be not, et a diameter bedrawn through the poliati contact, his diameter must also bisec the parallel to the an, gent: hicli is absurd .
36쪽
COR. 5. An a traight line drawn throughthe vertex os a diameter, oras to e parallel tostraight lines ordinatet applied to that diameter, touches the parabola This is manifeststo Cor. I.
I from a potnt of a parabola astraight line e drawn perpendicular o a diameter, and is froni the fame potnt a traight in boordinatet applied to that diameter the quare of the perpendicula is qua to the rectangle contained by the abscissa of the diameter and the latus reclum f
V. B. An abscissa is the segmen intercept-ed etwix the vertex o a diameter an astraight in ordinatet applied to that diameter.
37쪽
Casera Then the diameter is the axis os theparabola.
Let Die a poliat in a parabola, and DH a Fig. S. perpendicular to the axis BC H illi parallel to 5. l. the traight line to uch in theparabola in the verte of the axis and there- fore illo ordinately II. I. applied to the axis : dra DC to the focus, andi perpendicular to the directrix AB, and leti he the
to DA, that is to DC, the square of Hreis equalto the quare o DC, that is to the quare ofDH, together illi the quare of H C: ut, sincera is quai to C, the fame quare of HB is equa to seu times the rectan glem , together illi the 8. 2. Elem. square o HC: therei ore the quare of DH, together illithe quare of HC, is qua to seu times therectangle FB together illi the quare os HC there re the quare o Hreis equa tofour times the rectangle FB that is to therectangi containe by the abscissa HF, and the parameter of the Xis.
38쪽
Cas 2. When the diameter o hic the perpendicula is drawn is no the XiS. Fig. . . tela be perpendicular to the diameter n. l. a. AD; et ELie an ordinate to AD, and D theverteno the fame AD; the quare o E is equa to the rectangiae contained by the abscissa D and the parameter of the axis. Drarum parallel o LE; D vult there- fore 5 Cor. II. I. touch the parabola in D and let the famem meet the axis in K et Eribe drawn at right angies to the directrix and et a circleae describesseo the Centre E, distanc EF and this circle ill ouch cor. 16 3. Elem. the directri in F, an pass
Iet i meet the circumference of the Circle again
in II, and the stra ight lines DK, L in thepoinis P, 4 and leti me et the axis in o. Because the an gles 9. of this book, 4. I. Elem. CP and CB are right angies, and theangle BCP common the triangles CBA, PKare equiangular: AC, theresere, is q. 6. Elem.)
39쪽
that is to the rectangi containe by the abscissa LD, and the parameter of the axis. COR. 1. Henc the squares of perpendiculars drawn stom an potnts of a parabolito any diameters, are to ne l. 6. Elem. another, asthe abscissas intercepte between the vertices of thos diameter and the ordinates ra nfroin thos potntS.CoR. . The quare os straight lines ordi natet applied to the fame diameter, arest Oneanother, a the abscissas belween thos Straight
40쪽
lines and the vertex of that diameter LetEL, Q be ordinatet applied to the diameter D and let EN, Si perpendicula tothe fame hecause the triangle L is equiangula to the triangle QRS, the quare of ELis to that of QR a the quare o E to thatos QS, that is, by the precedin Corollary, Sthe abscissa DL to the abscissa DR.
COR. 3. Anxi hom the vertices of two diameter there e drawn traight lines ordinate-I applied to those two diameters that is, fili straight line drawn fro the verte of each diameteri an ordinate to the ther diameter, the abscissas etween hos ordinates and thet O Vertices are qua to ach other; for the perpendicular drawn seo the two vertices toth truo diameter are equat
Is rom a potnt of a parabola astraight line e drawn ordinatelyapplied to a diameter, the SqUare