An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page 2
... , we 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 ) . Then a will be Fig . 2 . B negative , while B , y are positive . Hence 2 MODERN GEOMETRY .
... , we 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 ) . Then a will be Fig . 2 . B negative , while B , y are positive . Hence 2 MODERN GEOMETRY .
Page 3
... Hence , twice the area PBC will be represented by aa , and we shall therefore have as before - aa + bB + cry = 2A . Thirdly , let P lie between AB , AC , produced backwards ( fig . 3 ) , so that B , 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 + cry = 2A . Thirdly , let P lie between AB , AC , produced backwards ( fig . 3 ) , so that B , y are negative while a is positive ...
Page 5
... Hence , r2 will be of the form - - -- 1 ( B1 — B2 ) ( Y2 − Y2 ) + m ( Y1 — Y2 ) ( α , − α2 ) + n ( α , — α2 ) ( ẞ1 — B2 ) , - - - dicular to AB , PN , QN ' to AC . Draw Qm perpendicular to PM , Qn to PN , and join mn . Then mn r = PQ ...
... Hence , r2 will be of the form - - -- 1 ( B1 — B2 ) ( Y2 − Y2 ) + m ( Y1 — Y2 ) ( α , − α2 ) + n ( α , — α2 ) ( ẞ1 — B2 ) , - - - dicular to AB , PN , QN ' to AC . Draw Qm perpendicular to PM , Qn to PN , and join mn . Then mn r = PQ ...
Page 6
... Hence 2A = ; 2A 2A a2 = -1 24 24 ; a2bc .. l 4A2 . Similarly ab2c m 4A2 , abc2 n 442 Hence r2 : abc - - 42 ( α ( B1 - B2 ) ( Y1 - Y2 ) + b ( Y1 — Y2 ) ( α , — α ) { a - - + c ( α1 − α2 ) ( ẞ2 − B1 ) } . This is one form of the ...
... Hence 2A = ; 2A 2A a2 = -1 24 24 ; a2bc .. l 4A2 . Similarly ab2c m 4A2 , abc2 n 442 Hence r2 : abc - - 42 ( α ( B1 - B2 ) ( Y1 - Y2 ) + b ( Y1 — Y2 ) ( α , — α ) { a - - + c ( α1 − α2 ) ( ẞ2 − B1 ) } . This is one form of the ...
Page 7
... Hence 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 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
a₁ aa+bB+cy Algebra angular points anharmonic ratio Arithmetic asymptotes auxiliary conic b₁ c₁ CAMBRIDGE CLASS BOOKS centre Chap chord cloth co-ordinates coefficients College common tangents condition conic passing conic section conics intersect Crown 8vo curve determine directrix ellipse find the equation fixed point focus four points given conic given point given straight line Hence hyperbola imaginary investigated line at infinity line joining locus meets the conic obtain pair parabola Pascal's Theorem perpendicular point f point of intersection points at infinity points of contact PQRS projection prove reciprocal polars represented respect right angles second degree Second Edition Similarly sin POS system of conics tangents tangents drawn theorem three points three straight lines touches the line Treatise triangle of reference U'bc ua² V'ca Vb² vß² W'ab wy² λα