Elementary Synthetic Geometry of the Point, Line and Circle in the Plane |
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Page 67
... touch a O only once . 3. A line which touches a O cannot cut it . 4. A is determined by two points if one of them is a given point of contact on a given line ; or , only one circle can pass through a given point and touch a given line ...
... touch a O only once . 3. A line which touches a O cannot cut it . 4. A is determined by two points if one of them is a given point of contact on a given line ; or , only one circle can pass through a given point and touch a given line ...
Page 69
... 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 contact when one circle lies within the other . Cor . 2. Since ...
... 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 contact when one circle lies within the other . Cor . 2. Since ...
Page 71
... touch the circle S at the points A , B , and C , and inter- sect to form the AA'B'C ' . O being the centre of the circle , LAOC = 24A'OC ' . C Proof .-- AC ' = BC ' , and Similarly BA ' = CA ' , ( 114 ° , Cor . 1 ) S AAOC ' = ABOC ...
... touch the circle S at the points A , B , and C , and inter- sect to form the AA'B'C ' . O being the centre of the circle , LAOC = 24A'OC ' . C Proof .-- AC ' = BC ' , and Similarly BA ' = CA ' , ( 114 ° , Cor . 1 ) S AAOC ' = ABOC ...
Page 74
... 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 P , Q , R , and S. 10. A variable chord passes through a ...
... 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 P , Q , R , and S. 10. A variable chord passes through a ...
Page 75
... touch one another , any line through the point of contact determines arcs which subtend equal angles in the two circles . 17. If any two lines be drawn through the point of contact of two touching circles , the lines determine arcs ...
... touch one another , any line through the point of contact determines arcs which subtend equal angles in the two circles . 17. If any two lines be drawn through the point of contact of two touching circles , the lines determine arcs ...
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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