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For, draw the diameter CHE, and the tangent TE, and its parallels PK, RI, MH, meeting the conjugate of the diameter CR in the points T, K, I, M. Then, because similar triangles are as the squares of their like sides, it is,
by sim. triangles, CR: GP2 :: A CRI: A GPK, CR2 GH2:: A CRI: A GHM;
theref. by division, CR2: GP2
Again, by sim. tri. CE: CH2:: A CTÉ: A CMH;
and by division,
CE2 CE2 CH :: А сСТЕ : ТЕНМ.
But, by cor. 5 theor. 19, the ACTE = ▲ CIR,
and by cor. I theor. 19, TEHG = KPHG, or TEHM KPHM; theref. by equ. CE: CE CH2: CR2: GP2- GH' or PH.HQ.
In like manner CE2: CE2 CH2:: cr2: рH. нq.
Q. E. D.
Corol. 1. In like manner, if any other line p'H'q', parallel to cr or to pq, meet PHQ; since the rectangles PH'Q, p'H'q' are also in the same ratio of CR to cr2; therefore rect. PHQ рHq PHQ: р'H'q.
Also, if another line pho' be drawn parallel to ra or CR; because the rectangles p'ha', p'hq are still in the same ratio, therefore, in general, the rect. PHQ : pнq:: P'ha' : p'hq'.
That is, the rectangles of the parts of two parallel lines, are to one another, as the rectangles of the parts of two other parallel lines, any where intersecting the former.
Corol. 2. And when any of the lines only touch the curve, instead of cutting it, the rectangles of such become squares, and the general property still attends them.
Corol. 3. And hence TE: Te tE: te.
OF THE HYPERBOLA.
THEOREM XIV (5).
The Sum or Difference of the Semi-transverse and a Line drawn from the Focus to any Point in the Curve, is equal to a Fourth Proportional to the Semi-transverse, the Distance from the Centre to the Focus, and the Distance from the Centre to the Ordinate belonging to that Point of the Curve.
For, draw AG parallel and equal to ca the semi-conjugate; and join CG meeting the ordinate DE produced in н.
but, by right angled triangles, FD2 + de2 = fe2 ;
therefore FE2 CF2 ca2
But by theor. 4, and, by supposition, theref. FE = CA2
2CF. CD+CD2 + DH2.
ca2 = CA2,
2CF. CD=2CA. CI;
CA2: CD2 :: CF2 or CA2 + AG2 : CI2;
But, by supposition, and, by sim As, therefore
is FE CI square
CA = AL
In the same manner is found ƒE = CI + CA = BI. Q. E. D. Corol. 1. Hence CH = CI is a 4th propor. to CA, CF, CD.
Corol. 2. And ƒE + FE = 2CH or 2c1; or FE, CH, ƒE are in continued arithmetical progression, the common difference being ca the semi-transverse.
Corol. 3. From the demonstration it appears, that DE2= DH2 - AG2 = DH' ca. Consequently DH is every where greater than DE; and so the asymptote CGH never meets the curve, though they be ever so far produced: but DH and DE approach nearer and nearer to a ratio of equality as they recede farther from the vertex, till at an infinite distance they become equal, and the asymptote is a tangent to the curve at an infinite distance from the vertex.
THEOREM XV (11).
If a Line be drawn from either Focus, l'erpendicular to á Tangent to any Point of the Curve; the Distance of their Intersection from the Centre will be equal to the Semitransverse Axis.
That is, if FP, fp be perpendicular to the tangent TPPр, then shall CP and cp be each equal to CA or CB.
For, through the point of contact E draw FE and ƒE, meeting FP produced in G. Then, the GEP FEP, being each equal to the Ep, and the angles at P being right, and the side PE being common, the two triangles GEP, FEP are equal in all respects, and so GE FE, and GP fp. Therefore, since FPFG, and Fc = Ff, and the angle at F common, the side CP will be = fG or AB, that is CPCA or CB. And in the same manner cp = CA or CB. Q. E. D. Corol. 1. A circle described on the transverse axis, as a diameter, will pass through the points P, p; because all the lines CA, CP, CP, CB, being equal, will be radii of the circle.
Corol: 2. CP is parallel to ƒE, and cp parallel to FE.
Corol. 3. If at the intersections of any tangent, with the circumscribed circle, perpendiculars to the tangent be drawn, they will meet the transverse axis in the two foci. That is, the perpendiculars PF, pf give the foci F, f.
THEOREM XVI (12).
The equal Ordinates, or the Ordinates at equal Distances from the Centre, on the opposite Sides and Ends of an Hyperbola, have their Extremities connected by one Right Line passing through the Centre, and that Line is bisected by the Centre.
That is, if CD = CG, or the ordinate DE = GH; then shall
CE CH, and ECH will be a right line.
For, when CD = CG, then also is DEGH by cor. 2 theor. 1. But the 4D=4G, being both right angles;
therefore the third side CE = CH, and the 4DCE GCH, and consequently ECH is a right line.
Coral. 1. And, conversely, if ECH be a right line passing through the centre; then shall it be bisected by the centre, or have CE = CH; also DE will be =GH, and CD = CG.
Corol. 2. Hence also, if two tangents be drawn to the two ends E, H of any diameter EH; they will be parallel to each other, and will cut the axis at equal angles, and at equal distances from the centre. For, the two CD, CA being equal to the two CG, CB, the third proportionals CT, cs will be equal also; then the two sides CE, CT being equal to the two CH, cs, and the included angle ECT equal to the included angle HCS, all the other corresponding parts are equal: and so the <T=2s, and TE parallel to HS.
Corol. 3. And hence the four tangents, at the four extremities of any two conjugate diameters, form a parallelogram inscribed between the hyperbolas, and the pairs of opposite sides are each equal to the corresponding parallel conjugate diameters. For, if the diameter eh be drawn parallel to the tangent TE or HS, it will be the conjugate to EH by the definition; and the tangents to eh will be parallel to each other, and to the diameter EH for the same reason.
THEOREM XVII (13).
If two Ordinates ED, ed be drawn from the Extremities E, e, of two Conjugate Diameters, and Tangents be drawn to the same Extremities, and meeting the Axis produced in T and R;
Then shall CD be a mean Proportional between cd, dr, and cd a mean Proportional between
cd. de, or ▲ CDE = A cde.
Also cd CD. DT, and CD2 = cd. dr. Or cd a mean proportional between CD, DT; and CD a mean proportional between cd, dr.
THEOREM XVIII (14).
Q. E. D.
The same Figure being constructed as in the last Proposition, each Ordinate will divide the Axis, and the Semi-axis added to the external Part, in the same Ratio.
Hence, and from cor. 3 to the last prop. it is,
cd2 = CD. DT = AD. DB = CD2
and cd.dR = Ad. dB = CA2
Corol. 2. Hence also CA CD2-cd2, and ca2=de2 — DE2.
Corol. 3. Farther, because CA2: ca2:: AD. DB or cď2: DE2.
ca: cd: DE.