An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page 22
... lying on BC , Q , Q ' on CA , R , R ' on AB ; BQ , CR meet AP ' in D ̧ D2 ; CR , AP meet BQ ' in E1 , E2 ; AP , BQ meet CR ' in F1 , F2 ; CD1 , BD , intersect in L ; AE , CE , in M ; BF1 , AF in N. Prove that AL , BM , CN intersect in a ...
... lying on BC , Q , Q ' on CA , R , R ' on AB ; BQ , CR meet AP ' in D ̧ D2 ; CR , AP meet BQ ' in E1 , E2 ; AP , BQ meet CR ' in F1 , F2 ; CD1 , BD , intersect in L ; AE , CE , in M ; BF1 , AF in N. Prove that AL , BM , CN intersect in a ...
Page 24
... appears also that , if four points P , Q , R , S , lying in a straight line , be joined with each of two other points O , O ' , the anharmonic ratios of the pencils OP , OQ , . OR , OS ; O'P , O'Q , O'R 24 TRILINEAR CO - ORDINATES .
... appears also that , if four points P , Q , R , S , lying in a straight line , be joined with each of two other points O , O ' , the anharmonic ratios of the pencils OP , OQ , . OR , OS ; O'P , O'Q , O'R 24 TRILINEAR CO - ORDINATES .
Page 25
... lies , is said to be divided harmonically . From what has been said above , it will be seen that , if an harmonic pencil be cut by à transversal , the four points of section will form an harmonic range . And if four points , forming an ...
... lies , is said to be divided harmonically . From what has been said above , it will be seen that , if an harmonic pencil be cut by à transversal , the four points of section will form an harmonic range . And if four points , forming an ...
Page 32
... lying in a straight line , the sine of the angle between any two lines taking the place of the mutual distance of two points . From the proposition , proved in Art . 20 , it will follow that , if a system of straight lines in involution ...
... lying in a straight line , the sine of the angle between any two lines taking the place of the mutual distance of two points . From the proposition , proved in Art . 20 , it will follow that , if a system of straight lines in involution ...
Page 46
... lies within the concavity of the conic . 14. Since the two tangents drawn through B meet the conic in points situated in the line CA , it follows that CA is the chord of contact of tangents to the conic drawn through B , or that CA is ...
... lies within the concavity of the conic . 14. Since the two tangents drawn through B meet the conic in points situated in the line CA , it follows that CA is the chord of contact of tangents to the conic drawn through B , or that CA is ...
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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