Elementary Synthetic Geometry of the Point, Line and Circle in the Plane |
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Page 66
... of a square , and A , E , F of an equilateral triangle inscribed in the same circle . What is the angle between the lines BE and DF ? between BF and ED ? SPECIAL SECANTS -- TANGENT . 109 ° . Let P 66 SYNTHETIC GEOMETRY .
... of a square , and A , E , F of an equilateral triangle inscribed in the same circle . What is the angle between the lines BE and DF ? between BF and ED ? SPECIAL SECANTS -- TANGENT . 109 ° . Let P 66 SYNTHETIC GEOMETRY .
Page 67
... tangent . Def . 1.- -A tangent to a circle is a secant in its limiting position when its points of intersection with the circle become coincident . That the tangent cannot cut or cross the O is evident . For if it cuts the O at P it ...
... tangent . Def . 1.- -A tangent to a circle is a secant in its limiting position when its points of intersection with the circle become coincident . That the tangent cannot cut or cross the O is evident . For if it cuts the O at P it ...
Page 68
... tangent to the same point on a circle are perpendicular to one another . ଓ T P S L ' is a centre - line and Ta tan ... tangent at the point of con- tact is a centre - line . ( Converse of the theorem . ) Cor . 3. The perpendicular to a ...
... tangent to the same point on a circle are perpendicular to one another . ଓ T P S L ' is a centre - line and Ta tan ... tangent at the point of con- tact is a centre - line . ( Converse of the theorem . ) Cor . 3. The perpendicular to a ...
Page 70
... tangent to S. Similarly BP is tangent to S. Cor . 1. Since PO is the right bisector of AB , PA = PB . ( 106 ° , Cor . 4 ) ( 110 ° , Cor . 3 ) ( 113 ° ) ( 53 ° ) Hence calling the segment PA the tangent from P to the circle , when length ...
... tangent to S. Similarly BP is tangent to S. Cor . 1. Since PO is the right bisector of AB , PA = PB . ( 106 ° , Cor . 4 ) ( 110 ° , Cor . 3 ) ( 113 ° ) ( 53 ° ) Hence calling the segment PA the tangent from P to the circle , when length ...
Page 71
... tangents at A and C are fixed , and the tangent at B is variable , we have the following theorem : - The segment of a variable tangent intercepted by two fixed tangents , all to the same circle , subtends a fixed angle at the centre ...
... tangents at A and C are fixed , and the tangent at B is variable , we have the following theorem : - The segment of a variable tangent intercepted by two fixed tangents , all to the same circle , subtends a fixed angle at the centre ...
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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