A geometrical treatise on conic sections, with numerous examplesFor the use of schools and students in the universitiesWith an appendix on harmonic ratio, poles and polars, and reciprocation . Draw the ordinate PN, and produce it to meet CD in K. Also draw Pn at right angles to CB, and let the tangentat P meet CB produced in t. Now since the angles at iVand .Fare right angles, it isevident that a circle may be described about the quadrilateralfigure NKFG; .-. PG .PF= PN. PK, {Euclid, III. 36 Cor.)= Ct . Cn,= BC\ (Prop. XIV.) 56 CONIC SECTIONS. Prop. XXIV. 41. If Pbe any point on the ellipse, a
A geometrical treatise on conic sections, with numerous examplesFor the use of schools and students in the universitiesWith an appendix on harmonic ratio, poles and polars, and reciprocation . Draw the ordinate PN, and produce it to meet CD in K. Also draw Pn at right angles to CB, and let the tangentat P meet CB produced in t. Now since the angles at iVand .Fare right angles, it isevident that a circle may be described about the quadrilateralfigure NKFG; .-. PG .PF= PN. PK, {Euclid, III. 36 Cor.)= Ct . Cn,= BC\ (Prop. XIV.) 56 CONIC SECTIONS. Prop. XXIV. 41. If Pbe any point on the ellipse, and CD be conjugateto OP, then SP . SP = L>raw the normal PG and produce it to meet CD in F;then since CD is parallel to the tangent at P, .-. PPis at right angles to CD, . CD = AC. BC, (Prop. XXII. Cor.) and PF. PG = B C2 = BC . B C, (Prop XXIII.) .-. CD: PG:: AC : BC. (1) Again, SP: SG :: CA : CS, (Prop. XL) SP: SG :: CA : 5P. SP: SG. SG:: CA : GS2, .. SP. &P: SP. SP- S£ . S# :: CM2 : C^l2 - <7S2. But SP. SP- SG . SG = PG2, (Euclid, VI. iVop. B) .-. SP. SP:PG2 :: CM2 :PC2. But from (1) CP2 : P672 :: CA2 : BC .-. SP. SP= CD2. This proposition may also be very easily deduced fromProp. XV. CONIC SECTIONS. 57 Prop. XXV. 42. The area of the ellipse is to the area of the auxiliarycircle as B G to A G
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
