An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page 1
... . Let P be the given point , and first suppose it to lie within F. 1 the triangle of reference ( fig . 1 ) . 831 CHAPTER TRILINEAR CO-ORDINATES EQUATION OF A STRAIGHT LINE ARTS DEFINITION of Trilinear Co-ordinates ARTS.
... . Let P be the given point , and first suppose it to lie within F. 1 the triangle of reference ( fig . 1 ) . 831 CHAPTER TRILINEAR CO-ORDINATES EQUATION OF A STRAIGHT LINE ARTS DEFINITION of Trilinear Co-ordinates ARTS.
Page 2
... aa + bB + cy = 2A . Next , suppose P to lie between AB , AC produced , and on the side of BC remote from A ( fig . 2 ) . Then a will be Fig . 2 . B E . AREA OF THE TRIANGLE OF REFERENCE . 3 negative , 2 TRILINEAR CO - ORDINATES .
... aa + bB + cy = 2A . Next , suppose P to lie between AB , AC produced , and on the side of BC remote from A ( fig . 2 ) . Then a will be Fig . 2 . B E . AREA OF THE TRIANGLE OF REFERENCE . 3 negative , 2 TRILINEAR CO - ORDINATES .
Page 12
... suppose the equation of the required straight line to be La + MB + Ny = 0 . We must then have Lf + Mg + Nh = 0 , whence L Lf ' + Mg ' + Nh ' = 0 ; = M N - gh ' - g'h hf ' - h'ƒ ̄ ̄ ̄fg ' — ƒ'g ' giving , as the equation of the required ...
... suppose the equation of the required straight line to be La + MB + Ny = 0 . We must then have Lf + Mg + Nh = 0 , whence L Lf ' + Mg ' + Nh ' = 0 ; = M N - gh ' - g'h hf ' - h'ƒ ̄ ̄ ̄fg ' — ƒ'g ' giving , as the equation of the required ...
Page 13
... suppose the equation of that line to be λα + β + γ = 0 . Then λ , μ , v must satisfy the following equations : λα + μβ + γ = 0 , λα + β + γ = 0 , 2 λα + μβ , + νγg = 0 , 3 whence , eliminating λ , u , v , by cross multiplication , the ...
... suppose the equation of that line to be λα + β + γ = 0 . Then λ , μ , v must satisfy the following equations : λα + μβ + γ = 0 , λα + β + γ = 0 , 2 λα + μβ , + νγg = 0 , 3 whence , eliminating λ , u , v , by cross multiplication , the ...
Page 30
... suppose . Then these points are said to form a system in involution . If K be a point such that OK2 = k , K is called a focus of the system . INVOLUTION . 31 If 2 be positive , there will 30 MODERN GEOMETRY . ON INVOLUTION Definitions PAGE.
... suppose . Then these points are said to form a system in involution . If K be a point such that OK2 = k , K is called a focus of the system . INVOLUTION . 31 If 2 be positive , there will 30 MODERN GEOMETRY . ON INVOLUTION Definitions PAGE.
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Common terms and phrases
aa+bB+cy angular points anharmonic ratio asymptotes b₁ b₂ centre Chap chord coefficients common tangents condition of tangency conic section Crown 8vo curve determine directrix distance escribed circles Fcap find the equation fixed point foci focus four points given conic given point given straight line harmonic pencil Hence imaginary internal bisectors investigated Let the equation line at infinity line joining locus meets the conic nine-point circle obtain opposite sides pair parabola parallel Pascal's Theorem perpendicular point f point of intersection points at infinity points of contact pole prove radical axis reciprocated with respect rectangular hyperbola represented right angles second degree sin POS tangents tangents drawn theorem three points three straight lines touches the line triangle of reference ua² V'ca values vß² W'ab whence wy² λα
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