Elementary Synthetic Geometry of the Point, Line and Circle in the Plane |
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Page 69
... circles separate until A and B coincide . Then the circles touch and OO ' passes through the point of contact . Def . - Two circles which touch one another have external contact when each circle lies without the other , and internal ...
... circles separate until A and B coincide . Then the circles touch and OO ' passes through the point of contact . Def . - Two circles which touch one another have external contact when each circle lies without the other , and internal ...
Page 71
... circle , subtends a fixed angle at the centre . Ex . 2. If four circles touch two and two externally , the points of contact are concyclic . Let A , B , C , D be the centres of the circles , and P , Q , R , S be the points of contact ...
... circle , subtends a fixed angle at the centre . Ex . 2. If four circles touch two and two externally , the points of contact are concyclic . Let A , B , C , D be the centres of the circles , and P , Q , R , S be the points of contact ...
Page 74
... circles , and d the distance between them , the circles touch when d = r ± r . 8. Give the conditions under which two circles have 4 , 3 , 2 , or I common tangent . 9. Prove Ex . 2 , 116 ° , by drawing common tangents to the circles at ...
... circles , and d the distance between them , the circles touch when d = r ± r . 8. Give the conditions under which two circles have 4 , 3 , 2 , or I common tangent . 9. Prove Ex . 2 , 116 ° , by drawing common tangents to the circles at ...
Page 75
... circle is perpendicular to the centre - line through that point . 20. Three circles touch each other externally at A , B , and C. The chords AB and AC of two of the circles meet the third circle in D and E. Prove that DE is a diameter ...
... circle is perpendicular to the centre - line through that point . 20. Three circles touch each other externally at A , B , and C. The chords AB and AC of two of the circles meet the third circle in D and E. Prove that DE is a diameter ...
Page 80
... circles . Then ABEC is the triangle required . Proof . - BEBA and CE = CD . Since the circles intersect in another ... touch ( 113 ° , Def . ) and the triangle takes the limiting form and becomes a line . When BC is greater than the ...
... circles . Then ABEC is the triangle required . Proof . - BEBA and CE = CD . Since the circles intersect in another ... touch ( 113 ° , Def . ) and the triangle takes the limiting form and becomes a line . When BC is greater than the ...
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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