An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory Projections |
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Page 15
... obtain the condition under the form ( mn ' — m'n ) a + ( nl ′ — n'l ) b + ( lm ' — l'm ) c = 0 ...... ( 4 ) . - - This is the necessary condition of parallelism , and is generally the most convenient form which can be employed . It is ...
... obtain the condition under the form ( mn ' — m'n ) a + ( nl ′ — n'l ) b + ( lm ' — l'm ) c = 0 ...... ( 4 ) . - - This is the necessary condition of parallelism , and is generally the most convenient form which can be employed . It is ...
Page 30
... obtain a demonstration of the statements made in Art . 6 ; that the points in which the external bisectors of each angle of a triangle respectively intersect the sides opposite to them , lie in the same straight line ; and that the ...
... obtain a demonstration of the statements made in Art . 6 ; that the points in which the external bisectors of each angle of a triangle respectively intersect the sides opposite to them , lie in the same straight line ; and that the ...
Page 36
... distance , its co - ordinates will satisfy the equation aa + bB + cy = 0 . We hence obtain the following equation : λ2a2 + μ2b2 + v2c2 — 2μvbc — 2vλca — 2λμab = 0 , - which is equivalent to ± ( λa ) * ± 36 TRILINEAR CO - ORDINATES .
... distance , its co - ordinates will satisfy the equation aa + bB + cy = 0 . We hence obtain the following equation : λ2a2 + μ2b2 + v2c2 — 2μvbc — 2vλca — 2λμab = 0 , - which is equivalent to ± ( λa ) * ± 36 TRILINEAR CO - ORDINATES .
Page 37
... obtained by eliminating a between the equations λβη + μιγα + ναβ = 0 , la + mB + ny = 0 , must be coincident . The equation which determines these is − XlBy + ( μy + vß ) ( mß + ny ) = 0 , and the condition that the two values of B : y ...
... obtained by eliminating a between the equations λβη + μιγα + ναβ = 0 , la + mB + ny = 0 , must be coincident . The equation which determines these is − XlBy + ( μy + vß ) ( mß + ny ) = 0 , and the condition that the two values of B : y ...
Page 39
... obtained by writing a = 0 in the above may be a perfect square . This requires that u'2 = vw , or u ' = ± ( vw ) 1 . Similarly , v ' = ± ( wu ) , w ' = ± ( uv ) + , are necessary conditions that the conic should touch the lines B = 0 ...
... obtained by writing a = 0 in the above may be a perfect square . This requires that u'2 = vw , or u ' = ± ( vw ) 1 . Similarly , v ' = ± ( wu ) , w ' = ± ( uv ) + , are necessary conditions that the conic should touch the lines B = 0 ...
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Common terms and phrases
aa+bB+cy angular points asymptotes auxiliary conic b₁ Brianchon's Theorem C₁ centre Chap chord co-ordinates coefficients common tangents conic passing conic section determinant directrix find the equation fixed point fixed straight line focus follows four points given conic given point given straight line Hence imaginary internal bisectors investigated Let the equation line at infinity locus meets the conic nine-point circle obtain opposite sides pair parabola Pascal's Theorem perpendicular point f point of intersection points at infinity points of contact polar reciprocal projection prove radical axis reciprocal curve reciprocated with respect rectangular hyperbola represented right angles second degree similar and similarly system of conics tangents drawn theorem three points three straight lines triangle of reference ua² uf+w'g+v'h Vb² vß² W'ab Wc² whence wy²