An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory Projections |
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Page 6
... values of l , m , n are independent of the positions of the points , the distance of which we wish to find , suppose these points to be B and C. Then α1 = 0 , B1 = 24 b , 1 = 0 , 2A = ; C α2 = 0 , B2 = 0 , Y2 2 2A 2A also r = a . Hence ...
... values of l , m , n are independent of the positions of the points , the distance of which we wish to find , suppose these points to be B and C. Then α1 = 0 , B1 = 24 b , 1 = 0 , 2A = ; C α2 = 0 , B2 = 0 , Y2 2 2A 2A also r = a . Hence ...
Page 12
... values of a , B , y , at the point of intersection . 10. To find the equation of the straight line , passing through two given points . Let f , g , h ; f , g , h , be the co - ordinates of the two given points , and suppose the equation ...
... values of a , B , y , at the point of intersection . 10. To find the equation of the straight line , passing through two given points . Let f , g , h ; f , g , h , be the co - ordinates of the two given points , and suppose the equation ...
Page 14
... values of α , β , γ . This gives , eliminating a , B , y by cross - multipli- cation , 1mn − 1 ̧m ̧n + l ̧m ̧n ̧ − Im‚n ̧ + l ̧m ̧n − 1 ̧m ̧n ̧ = 0 , the required condition . The identity of form between the conditions that three ...
... values of α , β , γ . This gives , eliminating a , B , y by cross - multipli- cation , 1mn − 1 ̧m ̧n + l ̧m ̧n ̧ − Im‚n ̧ + l ̧m ̧n − 1 ̧m ̧n ̧ = 0 , the required condition . The identity of form between the conditions that three ...
Page 16
... values of a , B , y , since , as we have already proved ( Art . 2 ) , aa + bB + cy = 24. Hence the equa- tion ( 4 ) may be looked upon as an expression of the fact that the two equations la + mẞ + ny = 0 , = l'a + m'B + n'y = 0 , cannot ...
... values of a , B , y , since , as we have already proved ( Art . 2 ) , aa + bB + cy = 24. Hence the equa- tion ( 4 ) may be looked upon as an expression of the fact that the two equations la + mẞ + ny = 0 , = l'a + m'B + n'y = 0 , cannot ...
Page 20
... - g r - g + ( If + mg + nh ) . h - զ h And from the values of q and r 1 q COS A r = = 1 b - cos A с - - − ( n — m cos A ) a ( l cos B + m cos A – n ) lbc α lbc 9 1 Similarly cos A r 1 1 + 2 20 TRILINEAR CO - ORDINATES .
... - g r - g + ( If + mg + nh ) . h - զ h And from the values of q and r 1 q COS A r = = 1 b - cos A с - - − ( n — m cos A ) a ( l cos B + m cos A – n ) lbc α lbc 9 1 Similarly cos A r 1 1 + 2 20 TRILINEAR CO - ORDINATES .
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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