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1. D eing aloint in a parabola, and DC Fig. 2.drawn rom D to the focus, and D drawn perpendicular to the directrix AB iE, that bisecis the angle CDA, ouches the parabola in the oin D. In DE ahe any the poliata; and havingjoined A, C AC, dra FG perpendicular to the directrix then, hecausem is equal I. I. Elem. to DC, DF Common, and the an-gle D equat o DC; C is equat 4. I. Elem. to FA; and consequently greater than FG there re the oin F is ithout the cor. 3. I. parabola; and consequently the
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Fig. 2. 2. ΗΚ, a traight in drawn through thevertex of the axis, and made perpendicula tothe axis, ouches the parabola. In HK ake
hecause, is greater 19 1 Elem. than CH, that is, than Cor. 1 1. HB, that is, than KL, KC is reater than KL: heres ore the polo Κis ithout the parabola, and ΗΚ ouches hieparabola. COR. 1. his proposition potnis out a methodos drawing a traight line that will ouch a parabola in a given mini, provide the directrixand the focus beatve in position. COR. 2. An since it has been proved thatali strat glit lines that ouch a parabola, fallwithout it toward the fame paris, that Curve, it is plain is every here convecon the ideon hicli theriouch inclines are, ut Concaveo the contrar fide.
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The directrix and the sociis of a parabola, an a traight sine notparallel o an diameter, einggiven in position to dra a
stra ight line parallel to the straight line give in position, hici, ill
toucti the Parabola. Ambeing the directri an C the secus o Fig. a parabola, and M a straight line not parallello any diameter it is required to dra astraight line parallelao MN, hicli,ill ouchthe parabola. Fro the focus ira C perpendiculario ΜΝ, and meeting the directri in A and havinibisected AC in E dra ED parallel o ΜΝ, and meeting the diameter through A in the oin D, and oin D; then, in the triangles ADE, DE is equat o CE, DCOmmon, and the angies atra right angies;
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h .'α. Q DOm a polrit E in a parabola, there e drawn a traight lino EG, ei ther parallel to the Xis, no bisecting the angi contained by the diameter passin throughthat oint, an a stra ight linedrawn rom the fame potnt to the focus ohe traight line EG cuts the parabola in ne ther Oint,
butio in more than One. From the focus C et a perpendicula bedra n to EG, and let i meet the directrix in
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bola. There are two cases. The ne is that in Fig. 2.whicli EG passes through the focus: ecauSeEC is qua to EF, and ach of the an gles ECA, EF a right angle there fore AC is 5 and 6. I. Elem. equa to AF and Consequently it is qua to M and acti of theangles ACe, Me is a right angi : C is, there- fore, equata es and there fore the potntis is in the 2. I. parabola. In the ther case, EG passes no through Fig. 4. the focus. Fro the centre E at the distance n. i. NEC, describe a circle, meetin C again in H; an describe another circle through thepoliat C, H, is then, ecauset is qua toEF, and that EF is a right angle the circle describe seo the centre E ouches l6. 3. Elem. the directrix in F there re the rectangle AH is qua to 36. 3. Elem. thesquare of AF, that is to the square of Afr
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there re M ouches the circle 37. 3. Uem. OCH; and the centre of this circle is l9. 3. Elem. in te; it is algo in GE, whichiisecis Hat right angle M it is, there fore, in the potnt ewhere se, E interfectiachither. C, there- fore, is equa to es and there fore, the oin eis in the 2. I. parabola. It is evident, that EG cut the parabola nowhere but in the potnis E, r, is pos Sible, Iet E cui it also in another potnt e an leter te irawn perpendicular to the directrix Fig. 4. AB a Circle then, described fro in the Centreet, D in a distanc eC, passes through H, and ouches the directrix in theioin f, at a distance rom then polia A, es orareater tha that of the poliat f seo A and the square os in beingequat to the rectangle AH, is equat to thesquare of A Whicli is absurd. Co . O ali the traight lines that can bedrawn rom an potiat os a parabola, ni One Can ouch the parabola the diameter through the pol ni fatis 4. 1. with in the para-yota and an other traight line, Xcept thal
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whichiisecis the angi contained by the diameter through the oint, and the traight linedra wn fro the oin to the focus, meet theparabola again in another potnt.
IDDom the focus, of a parabola, di ,
perpendicula CG e drawn to any traight line LG, meeting the directrix in A; f the segment of the perpendicula intercepted be- tween the focus and the stra ight line, is no greater man iis thersegmen intercepte between thestraight line and the directrix, that is, is CG be not greater than GA the traight line LG, CCOS-sarily, meet the parabola.
When the segments CG, GA are equat, it is plain, rom hat wai demonstrated in Prop. 6.
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that the strat glit line G ouches the parabola in the poliat here the diameter through A
to in and Do the sint A, and on ei therside of A, place in the directrix, AF, or Aci, Such, that the quare os ei ther may be qualto the rectangle AH an hau in describe da circle through the oinis , H, F, draW, through F, E perpendicular o AF, and let FE meeti in D: and the quare of AFie- in equa to the rectangle AH, AF ouches the circle in F and there re the centre of the circle is in Fri: ut a CH is bisected at right an gles by the traight line LG, the centre of the circle is likewis in L it is, heresere, in Ε, theloint, here E, G intersect achother: enc Einis equat o EF and of con- Sequence, the oin E is in the parabola. In like manner, is es rawn perpendicular to the directrix meetsi in e the oin His in the
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COR Hence an stra ight line passing througha poliat with in a parabola, meet the parabola. Case 1. I the traight line is a diameter, itis evident, rom Prop. 4. I. that i meet theparabola.
Case 2. When the traight line is not a diameter. et L pas through the oint L with- in the parabola ; it,il me et the Parabola.-Fro the focus ciet the traight line CG bedra n at right angies t LG, and et ii meet the directrix in A and oin C, A thenbecause the ointi is illi in the parabola astraight line drawn fromi perpendicular to the directriX, is reater 3. I. than C LA, theres ore, hicli is no les tha this perpendicular, is greater than C AG theres ore, is greater than 4T. I. Elem. GC and theresere LG meet the parabola. PROP. IX THEOR.
The angi containe by a diametero a parabola, and a traight line
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drawn rom the verte of that diameter to the focus, is bisected by the traight in that ouches the parabola in that VerteX.
Fig. 5. The angle ADC, contained by the diameter AD, and the traight Iine DC, is bisected by DE, a straight inenouching the parabola in thevertex re ser si bisecis no the angle ADC, it is possibi for omemther traight linet do it and this ther traight line also illtouch the parabola 5. I. Whicli is absurd. COR. 1. O the otheriand is ADt a di ameter, and Eretouch the parabola in the vertex D of AD, and is the angi ADEae equalto the angi CDE, DC passe through the fo-Cus or, i DC passes through the focus, DEtouching the parabola in D, and the angle ADE ein equa to the angle CDE; A is