Elementary Synthetic Geometry of the Point, Line and Circle in the Plane |
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Page 257
... equianharmonic pen- cil at every vertex , and a given pencil determines an equian- harmonic range on every transversal . Cor . 4. Since the sine of an angle is the same as the sine of its supplement ( 214 ° , 1 ) , any ray may be ...
... equianharmonic pen- cil at every vertex , and a given pencil determines an equian- harmonic range on every transversal . Cor . 4. Since the sine of an angle is the same as the sine of its supplement ( 214 ° , 1 ) , any ray may be ...
Page 258
... equianharmonic pencils have a pair of cor- responding rays in common , the remaining rays intersect collinearly . 306 ° . Theorem . — If two equianharmonic ranges have three pairs of corresponding points in perspective , the fourth ...
... equianharmonic pencils have a pair of cor- responding rays in common , the remaining rays intersect collinearly . 306 ° . Theorem . — If two equianharmonic ranges have three pairs of corresponding points in perspective , the fourth ...
Page 268
... equianharmonic , and having three rays in common the fourth rays must be in common , i.e. , they can differ only by a straight angle , and therefore O , P , Q are collinear . ( Being the first application of anharmonic ratios the work ...
... equianharmonic , and having three rays in common the fourth rays must be in common , i.e. , they can differ only by a straight angle , and therefore O , P , Q are collinear . ( Being the first application of anharmonic ratios the work ...
Page 269
... equianharmonic ranges ABCD , A'B'C'D ' , and PQRS have their axes concurrent at Y , and AA ' , BB ' , CC ' , DD ' concurrent at X , and AP , BQ , CR , DS con- current at Z , then A'P , B'Q , C'R OF ANHARMONIC PROPERTIES . 269.
... equianharmonic ranges ABCD , A'B'C'D ' , and PQRS have their axes concurrent at Y , and AA ' , BB ' , CC ' , DD ' concurrent at X , and AP , BQ , CR , DS con- current at Z , then A'P , B'Q , C'R OF ANHARMONIC PROPERTIES . 269.
Page 270
... equianharmonic with the range ABCD . 1. If O ' be any other point on the circle , the LA'OB ' = LA'O'B ' , LB'OC ' = LB'O'C ' , etc. , and the two pencils O.A'B'C'D ' and O ' . A'B'C'D ' are equi- anharmonic . .. four points on a circle ...
... equianharmonic with the range ABCD . 1. If O ' be any other point on the circle , the LA'OB ' = LA'O'B ' , LB'OC ' = LB'O'C ' , etc. , and the two pencils O.A'B'C'D ' and O ' . A'B'C'D ' are equi- anharmonic . .. four points on a circle ...
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Common terms and phrases
ABCD Algebra altitude becomes bisects c.p.-circles centre of similitude chord of contact circles touch circumcircle co-axal coincide collinear concurrent concurrent lines concyclic congruent cut the circle denote diagonals diameter divided double points end-points equal angles equianharmonic equilateral triangle excircles external bisector fixed point geometric given circles given line given point harmonic range Hence hexagram homographic homologous hypothenuse incircle internal angles inverse points isosceles joins LAOB line-segment locus median middle point nine-points circle opposite sides orthogonally pair parallel parallelogram passes pencil perpendicular perspective plane point of contact point of intersection polar reciprocal Proof quadrangle radical axis radical centre radii radius rectangle rectilinear figure regular polygon rhombus right angle right bisector rotation secant similar Similarly square straight angle symbol tangent tensor theorem Theorem.-The three circles transversal vertex vertices