AHC cuts them, the square of AF is equal to the (36. 3. Elem.) rectangle CAH, that is, to the square of Af: therefore AF is equal to Af; but FE, AL, fe, are parallels therefore LE is (2. 6. Elem.) equal to Le.

COR. 1. On the other hand, if a straight line Ee, terminated both ways by a parabola, be bisected by the diameter AL, it is parallel to the tangent which passes through D, the vertex of AL: for if the straight line touching the parabola in the point D, be not parallel to LE, let another straight line be drawn touching the parabola, and parallel to LE; then the diameter which passes through the point where this other straight line touches the parabola, bisects the straight line Ee: but, according to the hypothesis, the same Ee is bisected by the diameter AL: which is absurd.

COR. 2. All straight lines ordinately applied to any diameter, are parallel to one another.

COR. 3. If two or more parallels be terminated both ways by a parabola, the diameter which bisects the one, or one of them, bisects also the other, or the rest of them: for the one that is bisected by a diameter, is parallel to the straight line touching the parabola in the vertex of that diameter; and consequently the other, or the others, is, or are, parallel to the same straight line that touches the parabola in that vertex; and, therefore, is, or are, bisected by the same diameter.

COR. 4. Any straight line, on the contrary, which bisects two parallels terminated both ways by a parabola, is a diameter for if it is not, it is possible for some other

straight line bisecting one of the parallels to be a diameter; and being a diameter, this other straight line must also bisect the other of them: but, according to the hypothesis, the former of the straight lines bisects both the parallels: which is absurd. And if from the point of contact a straight line be drawn bisecting another straight line parallel to the tangent, and terminated both ways by the parabola, that straight line is a diameter: for if it be not, let a diameter be drawn through the point of contact, this diameter must also bisect the parallel to the tangent: which is absurd.

COR. 5. And a straight line drawn through the vertex of a diameter, so as to be parallel to straight lines ordinately applied to that diameter, touches the parabola. This is manifest from Cor. 1.

PROP. XII. THEOR.

If from a point of a parabola a straight line be drawn perpendicular to a diameter, and if from the same point a straight line be ordinately applied to that diameter; the square of the perpendicular is equal to the rectangle contained by the abscissa of the diameter and the latus rectum of the axis.

Case 1. When the diameter is the axis of the parabola.

Let D be a point in a parabola (fig. 5.) and DH perpendicular to the axis BC; DH will be parallel to (5. 1.) the straight line touching the parabola in the vertex of the axis; and therefore will be ordinately (11. 1.) applied to *the axis: draw DC to the focus, and DA perpendicular to

the directrix AB, and let F be the vertex of the axis : then, because HB is equal to DA, that is, to DC, the square of HB is equal to the square of DC, that is to the square of DH, together with the square of HC: but, since BF is equal to FC, the same square of HB is equal to four times the rectangle HFC, together with the (8.2. Elem.) square of HC: therefore the square of DH, together with the square of HC, is equal to four times the rectangle HFB, together with the square of HC: therefore the square of HD is equal to four times the rectangle HFB, that is, to the rectangle contained by the abscissa HF, and the parameter of the axis.

Case 2. When the diameter to which the perpendicu lar is drawn is not the axis.

Let EN (fig. 4. n. 1. 2.) be perpendicular to the diameter AD; let EL be an ordinate to AD, and D the vertex of the same AD; the square of EN is equal to the rectangle contained by the abscissa LD and the parame ter of the axis.

Draw DK parallel to LE; DK will therefore (5. cor, 11. 1.) touch the parabola in D: and let the same DK meet the axis in K; let EF be drawn at right angles to the directrix; and let a circle be described from the centre E, distance EF; and this circle will touch (cor. 16, 3. Elem.) the directrix in F, and pass through the focus C: let AC be joined, and let it meet the circumference of the circle again in H, and the straight lines DK, LE in the points P, G; and let LE meet the axis in O,

Because the angles (9. of this book, 4. 1. Elem.) CPK

and CBA are right angles, and the angle BCP common, the triangles CBA, CPK are equiangular: AC, therefore, is (4. 6. Elem.) to CB, as CK to CP, that is, as (2. 6.; 16. 5. Elem.) OK to GP: the rectangle, therefore, contained by CA, GP is equal to (16. 6. Elem.) that contained by OK, CB: but because CA is (9. of this book, and 4. 1. Elem.) the double of CP, and CH the double of CG, AH is double of GP; and, consequently, the rectangle CAH is equal to twice the rectangle CA, GP, that is, to twice the rectangle OK, CB: but the square of EN, or of AF, is equal (36. 3. Elem.) to the rectangle CAH: it is therefore equal to twice the rectangle OK, CB, that is, to the rectangle contained by the abscissa LD, and the parameter of the axis.

COR. 1. Hence the squares of perpendiculars drawn from any points of a parabola to any diameters, are to one (1.6. Elem.) another, as the abscissas intercepted between the vertices of those diameters and the ordinates drawn from those points.

COR. 2. The squares of straight lines ordinately applied to the same diameter, are to one another, as the abscissas between those straight lines and the vertex of that diameter. Let EL, QR be ordinately applied to the diameter DN; and let EN, QS be perpendicular to the same: because the triangle ELN is equiangular to the triangle QRS, the square of EL is to that of QR, as the square of EN to that of QS, that is, by the preceding corollary, as the abscissa DL to the abscissa DR.

COR. 3. And if from the vertices of two diameters there be drawn straight lines ordinately applied to those

two diameters, that is, if the straight line drawn from the vertex of each diameter be an ordinate to the other diameter, the abscissas between those ordinates and the two vertices are equal to each other; for the perpendiculars drawn from the two vertices to the two diameters are equal.

PROP. XIII. THEOR.

If from a point of a parabola a straight line be drawn ordinately applied to a diameter, the square of half that ordinate is equal to the rectangle contained by the abscissa between that same ordinate and the vertex of that diameter, and the latus rectum of the same diameter.

Let AB be the directrix of a parabola (fig. 4. n. 1. 2.) and AD a diameter, to which EL, drawn from the point E of the parabola, is ordinately applied; and through the vertex D of the diameter AD, draw DK parallel to EL; DK, of consequence, will touch the parabola: draw DM perpendicular to the axis, and from Q, the vertex of the axis, draw QR ordinately applied to the diameter DL; and, consequently, parallel to EL.

Since QR is equal to DK, its square is equal to (47. 1. Elem.) the squares of DM, MK; but the square of DM is, by the first case of the foregoing proposition, equal to four times the rectangle MQB: and since MQ is equal (10. 1.) to QK, the square of MK is equal to four times the square of MQ: therefore the square of QR is equal to four times the rectangle MQB, together with four