Elementary Synthetic Geometry of the Point, Line and Circle in the Plane |
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Page vii
... harmonic division , etc. , as applied to the line and circle ; and it is believed that a student who becomes acquainted with these geometric extensions in this their simpler form will be greatly assisted in the wider discussion of them ...
... harmonic division , etc. , as applied to the line and circle ; and it is believed that a student who becomes acquainted with these geometric extensions in this their simpler form will be greatly assisted in the wider discussion of them ...
Page x
... Harmonic Ratio . SECTION III . —Anharmonic Properties . SECTION IV . — Polar Reciprocals and Reciprocation . SECTION V. - Homography and Involution , PAGE 178 252 PART I. GENERAL CONSIDERATIONS . 1o . A statement which X CONTENTS .
... Harmonic Ratio . SECTION III . —Anharmonic Properties . SECTION IV . — Polar Reciprocals and Reciprocation . SECTION V. - Homography and Involution , PAGE 178 252 PART I. GENERAL CONSIDERATIONS . 1o . A statement which X CONTENTS .
Page 252
Nathan Fellowes Dupuis. PART V. ON HARMONIC AND ANHARMONIC RATIOS— HOMOGRAPHY , INVOLUTION , ETC. SECTION I. GENERAL CONSIDERATIONS IN REGARD TO HARMONIC AND ANHARMONIC DIVISION . A D 298 ° . Let C be a point dividing a segment AB . The ...
Nathan Fellowes Dupuis. PART V. ON HARMONIC AND ANHARMONIC RATIOS— HOMOGRAPHY , INVOLUTION , ETC. SECTION I. GENERAL CONSIDERATIONS IN REGARD TO HARMONIC AND ANHARMONIC DIVISION . A D 298 ° . Let C be a point dividing a segment AB . The ...
Page 253
... anharmonic ratios is ± 1 , these ratios are equal . 4. Let AC . BD = +1 . AD . BC Then = AC AD and C and D are BC BD ' both internal or both external . AC - BC AD - BD AB AB Also = HARMONIC AND ANHARMONIC DIVISION . 253.
... anharmonic ratios is ± 1 , these ratios are equal . 4. Let AC . BD = +1 . AD . BC Then = AC AD and C and D are BC BD ' both internal or both external . AC - BC AD - BD AB AB Also = HARMONIC AND ANHARMONIC DIVISION . 253.
Page 254
... harmonic . ( 208 ° , Cor . 1 ) Harmonic division and harmonic ratio have been long em- ployed , and from being only a special case of the more general ratio , this latter was named " anharmonic " by Chasles , " who was the first to ...
... harmonic . ( 208 ° , Cor . 1 ) Harmonic division and harmonic ratio have been long em- ployed , and from being only a special case of the more general ratio , this latter was named " anharmonic " by Chasles , " who was the first to ...
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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