A Treatise on the Geometry of the Circle and Some Extensions to Conic Sections by the Method of Reciprocation: With Numerous Examples |
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Page 2
... subtends at any fifth point O on the circle , this equation reduces to sin BOC sin AOD + sin COA sin BOD + sin AOB sinCOD = 0 , a result which is therefore true for any pencil of four lines , and is deduced directly from Ptolemy's ...
... subtends at any fifth point O on the circle , this equation reduces to sin BOC sin AOD + sin COA sin BOD + sin AOB sinCOD = 0 , a result which is therefore true for any pencil of four lines , and is deduced directly from Ptolemy's ...
Page 18
... subtends a right angle at every point on the string . ] 2. A closed curve of given perimeter is of greatest area when its form is a circle . A M N B [ Let A be any point on the curve , and take B such that AMB and ANB are equal in ...
... subtends a right angle at every point on the string . ] 2. A closed curve of given perimeter is of greatest area when its form is a circle . A M N B [ Let A be any point on the curve , and take B such that AMB and ANB are equal in ...
Page 25
... subtends a con- stant angle a at the centre of the circle ( 2α = π- -C ) . B Bi P 0 P A 1 1 1 Hence the variable triangle AB10 has a constant vertical angle ( a ) and altitude ( y ) , and therefore its base and area are minima when 4,0 ...
... subtends a con- stant angle a at the centre of the circle ( 2α = π- -C ) . B Bi P 0 P A 1 1 1 Hence the variable triangle AB10 has a constant vertical angle ( a ) and altitude ( y ) , and therefore its base and area are minima when 4,0 ...
Page 31
... subtended by the sides of the given one at the point . Now since A1 = ax , A = by , and A , = cz , 2 x2 + y2 + z2 : + + a2 and is consequently a minimum when Δι_ Δε since A1 + A2 + A3 = const . From ( 2 ) it is obvious that x = Y = У C ...
... subtended by the sides of the given one at the point . Now since A1 = ax , A = by , and A , = cz , 2 x2 + y2 + z2 : + + a2 and is consequently a minimum when Δι_ Δε since A1 + A2 + A3 = const . From ( 2 ) it is obvious that x = Y = У C ...
Page 38
... subtended at the centre of the circle by the sides a , b , c be denoted by 2a , 28 , 2y , this relation may b2 - a2 OC be written which is easily reducible to = c2 - ba OB ' cos 2a + cos 2y = 1 , or the projection XY of the intercept is ...
... subtended at the centre of the circle by the sides a , b , c be denoted by 2a , 28 , 2y , this relation may b2 - a2 OC be written which is easily reducible to = c2 - ba OB ' cos 2a + cos 2y = 1 , or the projection XY of the intercept is ...
Contents
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Other editions - View all
A Treatise on the Geometry of the Circle: And Some Extensions to Conic ... William J. M'Clelland No preview available - 2017 |
A Treatise on the Geometry of the Circle and Some Extensions to Conic ... William J. M'Clelland No preview available - 2018 |
Common terms and phrases
ABCD bisected bisector Brocard points centre of similitude centres of perspective circum-centre circum-circle coaxal system coincide common tangents concurrent concurrent lines conic constant corresponding cyclic polygon cyclic quadrilateral denote diagonals diameter distances equal equation equianharmonic ex-circles fixed point given circles hence hexagon in-circle infinity inscribed inverse points Isogonal Conjugates isosceles limiting points line at infinity lines joining locus maximum mean centre median median triangle meet middle point minimum multiples opposite sides orthocentre orthogonally pair of inverse parallel pedal triangle perpendiculars point of intersection points of contact polar polygon prove r₁ radical axis radius reciprocal rectangle right angles segments sides of ABC similar triangles Similarly Simson lines species squares subtend symmedian point system of points tangents Theorem transversal triangle ABC triangle PQR vertex vertical angle
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