An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory Projections |
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Page 2
... get aa + bB + cy = 2A . Next , suppose P to lie between AB , AC produced , and on the side of BC remote from A ( fig . 2 ) . Fig . 2 . Then a will be B E negative , while ẞ , y are positive . Hence 2 TRILINEAR CO - ORDINATES .
... get aa + bB + cy = 2A . Next , suppose P to lie between AB , AC produced , and on the side of BC remote from A ( fig . 2 ) . Fig . 2 . Then a will be B E negative , while ẞ , y are positive . Hence 2 TRILINEAR CO - ORDINATES .
Page 3
... Hence , twice the area PBC will be represented by aa , and we shall therefore have as before aa + bB + cy = 2A . Thirdly , let P lie between AB , AC , produced backwards ( fig . 3 ) , so that ß , y are negative while a is positive ...
... Hence , twice the area PBC will be represented by aa , and we shall therefore have as before aa + bB + cy = 2A . Thirdly , let P lie between AB , AC , produced backwards ( fig . 3 ) , so that ß , y are negative while a is positive ...
Page 5
... Hence , r2 will be of the form - - - - 1 ( B 、— B2 ) ( Y2 — Y2 ) + ̧m ( Y1 − Ÿ¿ ) ( α ̧ — α2 ) + n ( α , − α2 ) ( B1 – B2 ) , - dicular to AB , PN , QN ' to AC . Draw Qm perpendicular to PM , Qn to PN , and join mn . Then M " r = PQ ...
... Hence , r2 will be of the form - - - - 1 ( B 、— B2 ) ( Y2 — Y2 ) + ̧m ( Y1 − Ÿ¿ ) ( α ̧ — α2 ) + n ( α , − α2 ) ( B1 – B2 ) , - dicular to AB , PN , QN ' to AC . Draw Qm perpendicular to PM , Qn to PN , and join mn . Then M " r = PQ ...
Page 6
... Hence a2 = - -1 a2bc 442 . Similarly m = ab2c 442 , abc2 n 442 Hence r2 : = abc 4A2 ― - { a ( B1 — B2 ) ( Y1 — Y2 ) + b ( Y2 — ~ 2 ) ( α4 — α2 ) - + c ( α , - α ) ( B1 - B1 ) } . This is one form of the expression for r2 . It may also ...
... Hence a2 = - -1 a2bc 442 . Similarly m = ab2c 442 , abc2 n 442 Hence r2 : = abc 4A2 ― - { a ( B1 — B2 ) ( Y1 — Y2 ) + b ( Y2 — ~ 2 ) ( α4 — α2 ) - + c ( α , - α ) ( B1 - B1 ) } . This is one form of the expression for r2 . It may also ...
Page 7
... Hence or PG.AC - PH . AB , bB = cy . This is a relation between the co - ordinates of any point on the line AD , it therefore is the equation of that line . COR . It hence may be proved that the three straight lines , drawn through the ...
... Hence or PG.AC - PH . AB , bB = cy . This is a relation between the co - ordinates of any point on the line AD , it therefore is the equation of that line . COR . It hence may be proved that the three straight lines , drawn through the ...
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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