A Syllabus of Modern Plane Geometry |
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Page 9
... termed " isogonal conjugates , " and the rectangles under their perpendiculars on the three sides are equal . The symmedian point K has its perpendicular distances from the sides respectively proportional to those sides , and the sum of ...
... termed " isogonal conjugates , " and the rectangles under their perpendiculars on the three sides are equal . The symmedian point K has its perpendicular distances from the sides respectively proportional to those sides , and the sum of ...
Page 10
... ) DD ' : EE ' : FF ' a3 : 63 : c3 , and this circle is consequently termed the Triplicate - ratio circle . ( 3 ) The triangles FDE , E'F'D ' are equal to one another , and are each similar to ABC . ( 4 ) If Q , Q ' be the IO A SYLLABUS OF.
... ) DD ' : EE ' : FF ' a3 : 63 : c3 , and this circle is consequently termed the Triplicate - ratio circle . ( 3 ) The triangles FDE , E'F'D ' are equal to one another , and are each similar to ABC . ( 4 ) If Q , Q ' be the IO A SYLLABUS OF.
Page 11
... termed the Brocard circle . The points Q , Q ′ lie upon the Brocard circle ; and the angle noK = Ωσ Κ - 20 . COSINE CIRCLE . If through the symmedian point K be drawn the antiparallels ƒKế , dKƒ ' , eKď to BC , CA , AB respectively ...
... termed the Brocard circle . The points Q , Q ′ lie upon the Brocard circle ; and the angle noK = Ωσ Κ - 20 . COSINE CIRCLE . If through the symmedian point K be drawn the antiparallels ƒKế , dKƒ ' , eKď to BC , CA , AB respectively ...
Page 12
... termed a " Tucker circle , " whose centre bisects the distance between the circumcentres of ABC , A'B'C ' ( collinear with K ) . ( 2 ) The triangles ZXY , Y'Z'X ' are equal to each other , and are each similar to ABC . The circum ...
... termed a " Tucker circle , " whose centre bisects the distance between the circumcentres of ABC , A'B'C ' ( collinear with K ) . ( 2 ) The triangles ZXY , Y'Z'X ' are equal to each other , and are each similar to ABC . The circum ...
Page 16
... termed the " principle of duality . " Each of two descriptive theorems so correlated is said to be the dual of the other , and it will be found that if any descriptive property is demonstrated , its dual also holds . 3 The middle points ...
... termed the " principle of duality . " Each of two descriptive theorems so correlated is said to be the dual of the other , and it will be found that if any descriptive property is demonstrated , its dual also holds . 3 The middle points ...
Common terms and phrases
ABCD angular points antiparallel axis of projection bisect Brocard circle Brocard point centre of projection centres of similitude circumcircle co-axal circles coincident collinear points complete quadrangle complete quadrilateral concurrent lines conic conjugate points conjugates with respect COR.-The corresponding points cross ratio DEFINITIONS diagonal triangle equi-cross external bisectors figure focoids form a harmonic four points four straight lines given circle harmonic conjugates harmonic pencil harmonic range imaginary points infinite distance inscribed isogonal conjugates joining the centres limiting points line at infinity line be drawn locus meet middle points nine-point circle opposite sides opposite vertices orthocentre pair of conjugate perpendicular points lie points of intersection points or rays POLE AND POLAR polygons radical axis radii range or pencil right angles segments semiperimeter sides BC sponding straight line joining symmedian point system of co-axal tangents drawn Theor THEOREM three circles transversal vanishing line vertex
Popular passages
Page 12 - D are said to be harmonic conjugates of each other with respect to the points A and B, and AB is said to be harmonically divided by the points C and D. If C and D are harmonic with respect to A and B, then A and B are harmonic with respect to C and D. Harmonic range The four points A, B, C, D are referred to as a harmonic range, denoted by (ABC D), \fC and D are harmonic conjugates with respect to A and B.
Page 7 - The bisectors of the angles of a triangle meet the opposite sides in three points on a straight line.
Page 19 - The locus of a point from which tangents drawn to two given circles are equal is a straight line*.
Page 20 - The locus of a point whose powers with respect to two given circles are equal is called the radical axis of the two given circles.
Page 6 - DEFINITION. Each of the three straight lines which join the angular points of a triangle to the middle points of the opposite sides is called a Median of the triangle. ON PROP. 37. 1. If, in the figure of Prop. 37, AC and BD intersect in K, shew that (i) the triangles AKB, DKC are equal in area, (ii) the quadrilaterals EBKA, FCKD are equal. 2. In the figure of I. 16, shew that the triangles ABC, FBC are equal in area. 3. On the...