An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page 43
... chap . I. 10. To find the equations of the four circles which touch the three sides of the triangle of reference . These may be obtained most readily by the employment of the equations for the determination of the centre , obtained in ...
... chap . I. 10. To find the equations of the four circles which touch the three sides of the triangle of reference . These may be obtained most readily by the employment of the equations for the determination of the centre , obtained in ...
Page 70
... , then AF , BD , CE become the tangents at A , B , C , respectively , and we obtain the theorem enunciated in Ex . 1 , Chap . II . Again , by supposing D to coincide with B , and E with C , we 70 TRILINEAR CO - ORDINATES .
... , then AF , BD , CE become the tangents at A , B , C , respectively , and we obtain the theorem enunciated in Ex . 1 , Chap . II . Again , by supposing D to coincide with B , and E with C , we 70 TRILINEAR CO - ORDINATES .
Page 74
... Chap . II . ) , represents a conic section . 2. To find the point in which a straight line , drawn in a given direction through a given point of the conic , meets the conic again . Let f , g , h be the co - ordinates of the given point ...
... Chap . II . ) , represents a conic section . 2. To find the point in which a straight line , drawn in a given direction through a given point of the conic , meets the conic again . Let f , g , h be the co - ordinates of the given point ...
Page 78
... Chap . III . to be v ' = 0 u , w ' , w ' , v , u ' v ' , u ' , w or uvw + 2u'v'w ' — uu'2 — vv'2 — ww'2 = 0 . - - - 8. To find the equation of the polar of a given point . If through a given point any straight line be drawn cutting a ...
... Chap . III . to be v ' = 0 u , w ' , w ' , v , u ' v ' , u ' , w or uvw + 2u'v'w ' — uu'2 — vv'2 — ww'2 = 0 . - - - 8. To find the equation of the polar of a given point . If through a given point any straight line be drawn cutting a ...
Page 79
... Chap . I. ) . Again , the equations of the tangents at ( f1 , 91 , h1 ) , ( f2 , 92 , h2 ) respectively , are 2 ' 1 ƒ1 ( ua + w'ß + v'y ) +9 , ( w'a + vß + · u'y ) + h ̧ ( v'a + u'ß + wy ) = 0 , f2 ( ux + w'ß + v'g ) + 92 ( w'a + vß + ...
... Chap . I. ) . Again , the equations of the tangents at ( f1 , 91 , h1 ) , ( f2 , 92 , h2 ) respectively , are 2 ' 1 ƒ1 ( ua + w'ß + v'y ) +9 , ( w'a + vß + · u'y ) + h ̧ ( v'a + u'ß + wy ) = 0 , f2 ( ux + w'ß + v'g ) + 92 ( w'a + vß + ...
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Common terms and phrases
angular points anharmonic ratio asymptotes auxiliary conic b₁ b₂ C₁ centre Chap chord co-ordinates coefficients common tangents conic section determinant directrix find the equation fixed point fixed straight line follows four points given conic given point given straight line given triangle Hence investigated Let the equation line at infinity line joining locus meets the conic nine-point circle obtain opposite sides parabola Pascal's Theorem perpendicular point f points at infinity points of contact points of intersection PQRS prove radical axis reciprocal curve reciprocated with respect rectangular hyperbola represented right angles second degree shewn similar and similarly sin POS tangents drawn term a,b,c theorem three points three straight lines triangle of reference ua² uf+w'g+v'h ux² V'ca va² Vb² vß² W'ab whence wy² λα