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DG in K, and CT in X; the rectangle CHK is, in both figures, equal to the rectangle DHG, and, in fig. 2, the rectangle CTX is erțual to the sum of the squares of Cð and CT.

Part 1. The rectangle CHK in fig. 1 and 2, is equal to the rectangle HG.

For the rectangle HCK is equal to the square of CG (44. 1 Sup. and 17. 6 Eu.), therefore, in the ellipse, fig. 1, taking from each the square of CH, the excess of the rectangle HCK

above the square of CH, or, (3. 2 Eu.), the rectangle CHK, is equal to the ex ess of the square of CG above that of CH, or (5. 2 Eu.), the rectangle DHG; and, in the hyperbola, fig. 2, taking these equals from the square of CH, the excess of the square of CH above the rectangle HCK, or (2. 2 Eu.), the rectangle CHK, is equal to the excess of the square of CH above that of CG, or (6. 2 Eu.), the rectangle DHG.

Part 2. In fig. 2, the rectangle CTX is equal to the sum of the squares of CO and CT.

For the rectangle TCX is equal to the square of CO (44. I Sup. and 17. 6 Eu.), adding to each the square of CT, the rectangle TCX with the square of CT, or (3. 2 Eu.), the rectangle CTX is equal to the sum of the squares of CO and CT.

PROP. XLVI. THEOR.

The same things being supposed ; the rectangle under the segments

of the diameter, between the tangent and centre, and between the tangent and ordinate, is, in the case of an ellipse, or transverse diameter of a hyperbola, equal to the rectangle under the segments of the same, between the tangent and its vertices; and, in the case of a second diameter of a hyperbola, to the sum of the squares of the second semidiameter, and the segment of the same diameter, between the centre and tangent,

Part. 1. In fig. 1 and 2 of the 44th prop. the rectangle CKH is equal to the rectangle DKG.

For the rectangle HCK is equal to the square of CG (44. 1 Sup. and 17. 6 Eu.), therefore, in the ellipse, fig. 1, taking each from the square of CK, the excess of the square of CK above the rectangle HCK, or (2. 2. Eu.), the rectangle CKH, is equal to the excess of the square of CK above that of CG, or (6. 2 Eu.), the rectangle DKG ; and, in the hyperbola, fig. 2, taking

from these equals, the square of CK, the excess of the rectangle HCK above the square of CK, or (3. 2 Eu.), the rectangle CKH, is equal to the excess of the square of CG above that of CK, or (5.2 Eu.), the rectangle DKG.

Part 2. In fig. 2, the rectangle CXT is equal to the sum of the squares of Có and CX.

For the rectangle TCX is equal to the square of CO (44. 1 Sup. and 17. 6 Eu.), adding to each the square of CX, the rectangle TCX with the square of CX, or (3. 2 Eu.), the rectangle CXT, is equal to the sum of the squares of Co and CX.

PROP. XLVII. THEOR.

If two parallel right lines ( DR and GL, see fig. 1 and 2 to prop.

44), touching an ellipse or opposite hyperbolas, meet another tangent (RLK) ; the rectangle under the segments (DR and GL) of the parallels, between their contacts and the tangent which they meet, is equal to the square of the semidiameter (CO), to which they are parallel. And the rectangle (RPL), under the segments of the tangent (RL), which the parallels meet, between its contact (P), and the parallel tangents, is equal to the square of the semidiameter (CZ), which is parallel to it; as is the rectangle ( XPK), under the segments of any tangent (RP), meeting two conjugate diameters (co and DG), between the contact (P) and the diameters.

Part 1. The rectangle under DR and GL is equal to the

square of co.

The right line DG joining the contacts D and G is a diameter, for if G were not the other vertex of the diameter passing through D, a right line drawn from G parallel to DR, to the diameter passing through D, would meet that diameter within the section, for it is parallel to the tangent drawn through the vertex of the diameter remote from D (30. 1 Sup. and 30. 1 Eu.), which tangent falling wholly without the section (Def. 10. 1 Sup.), if the right line so drawn from G, did not meet that diameter within the section, it would meet the tangent drawn through the vertex remote from D, contrary to the definition of parallel right lines; therefore if DG were not a diameter, a right line drawn through G parallel to DR would enter the section and not be a tangent, contrary to the supposition.

And the diameter Co is conjugate to DG (Def. 14. 1 Sup.), and, if, in the ellipse, RL be parallel to DG, the proposition, as far as relates to the rectangle under DR and GL and the rectangle RPL is manifest.

But if RL be not parallel to DG in fig. 1, let RL, in fig. 1 and 2, meet DG in K, and CO produced in X, and let ordinates PH and PT be drawn to the diameters DG and CO (36. 1 Sup).

And because the rectangle DKG is equal to the rectangle CKH (46. 1 Sup.), DK is to CK, as HK is to GK (16. 6 Eu.), therefore, because of the parallels, DR is to CX, as HP to GL (4. 6 and 16. 5 Ew.), and therefore the rectangle under DR and GL is equal to the rectangle under CX and HP or CT (16.6 Eu.), or, which is equal (44. 1 Sup. and 17. 6 Eu.), the square of Co.

Part 2. The rectangle RPL is equal to the square of CZ.

Because DR is to GL, as RP is to PL (Cor. 2. 14. 1 Sup. and 22. 6 Eu.), the rectangle under DR and GL is similar to the rectangle RPL, and is therefore to that rectangle, as the square of DR is to the square of RP (22. 6 Eu.), or which is equal (42. 1 Sup.), as the square of co to the square of CZ ; whence, the rectangle under DR and GL being equal to the square of Co, by the preceding part, the rectangle RPL is equal to the square of CZ (14. 5 Eu).

Part 3. “The rectangle XPK is equal to the square of CZ.

Because the rectangle CHK is equal to the rectangle DHG (45. 1. Sup.), CH is to HG, as DH is to HK (16. 6 Eu.), and therefore, because of the parallels, XP is to PL, as RP is to PK (Cor. 2. 10. 6 and 11. 5 Eu.), and so the rectangle XPK is equal to the rectangle RPL (16. 6 Eu.), or, by the preceding part, to the square of CZ.

Cor. 1. In ellipses and hyperboles, a right line joining the contacts of two parallel tangents is a diameter, the right line DG, joining the contacts of the parallel tangents DR and GL, being in the demonstration of the 1st part of this prop. proved to be a diameter.

Cor. 2. If the right line XPK touching an ellipse, or hyperbola, meet two diameters CO and DG, and the rectangle XPK be equal to the square of the semidiameter CZ, conjugate to that which passes through the contact P; the diameters Co and DG are conjugate ones.

For if any other semidiameter, except CG, were conjugate to CO, the rectangle under XP, and a right line greater or less than PK, would be equal to the square of cz (part 3 of this prop.), contrary to the supposition.

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equal to KG; in all the cases draw through H the right line PHQ ordinately applied to the diameter HK, meeting the section or sections in P and Q; KP and KQ being joined touch the section or sections in P and Q; for if either of them, as KP, were not a tangent to the section, let a tangent PZ, drawn from P by part 1, meet the diameter HK in Z;

and in the case of an ellipse or hyperbola, CH, CG and CZ would be continually proportional (44. 1 Sup.), and therefore CG would have the same ratio to CK and čz (Constr. and 11. 5 Eu.), which is absurd (8. 5 Eu.); and in the case of a parabola, fig. 4, GZ would be equal to GH (44. 1 Sup.), or its equal by construction GK, which is also absurd (Ax. 9. 1 Eu).

Scholium. It appears from the construction of this problem, that two tangents may be drawn to a conick section or opposite sections, from any point without it or them, as the case may be, which is not in the asymptote of a hyperbola.

PROP. XLIX. THEOR.

A right line (KH), passing through the concourse (K) of two right

lines (PK and QK) touching a conick section or opposite sections, and bisecting the right line (PQ), joining their contacts, is a diameter of the section.

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For if KH be not a diameter, let a diameter HZ be drawn through H (Cor. 35. 1 Sup.), meeting the tangent PK in Z, and let QZ be drawn, meeting the section in D, through D let the right line DG be drawn parallel to PQ, meeting, in the case of fig. 1, because D is not the vertex of the diameter HZ, and DG

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