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In geometry, a set of Johnson circles comprises three circles of equal radius ''r'' sharing one common point of intersection ''H''. In such a configuration the circles usually have a total of four intersections (points where at least two of them meet): the common point ''H'' that they all share, and for each of the three pairs of circles one more intersection point (referred here as their 2-wise intersection). If any two of the circles happen to just touch tangentially they only have ''H'' as a common point, and it will then be considered that ''H'' be their 2-wise intersection as well; if they should coincide we declare their 2-wise intersection be the point diametrically opposite ''H''. The three 2-wise intersection points define the reference triangle of the figure. The concept is named after Roger Arthur Johnson.〔Roger Arthur Johnson, ''Modern Geometry'': ''An Elementary Treatise on the Geometry of the Triangle and the Circle'', Houghton, Mifflin Company, 1929〕〔Roger Arthur Johnson, "A Circle Theorem", ''American Mathematical Monthly 23, 161–162, 1916.〕〔(Roger Arthur Johnson (1890–1954) )〕 == Properties == # The centers of the Johnson circles lie on a circle of the same radius ''r'' as the Johnson circles centered at ''H''. These centers form the Johnson triangle. # The circle centered at ''H'' with radius 2''r'', known as the anticomplementary circle is tangent to each of the Johnson circles. The three tangent points are reflections of point ''H'' about the vertices of the Johnson triangle. # The points of tangency between the Johnson circles and the anticomplementary circle form another triangle, called the anticomplementary triangle of the reference triangle. It is similar to the Johnson triangle, and is homothetic by a factor 2 centered at ''H'', their common circumcenter. # Johnson's theorem: The 2-wise intersection points of the Johnson circles (vertices of the reference triangle ''ABC'') lie on a circle of the same radius ''r'' as the Johnson circles. This property is also well known in Romania as The 5 lei coin problem of Gheoghe Ţiţeica. # The reference triangle is in fact congruent to the Johnson triangle, and is homothetic to it by a factor −1. # The point ''H'' is the orthocenter of the reference triangle and the circumcenter of the Johnson triangle. # The homothetic center of the Johnson triangle and the reference triangle is their common nine-point center. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Johnson circles」の詳細全文を読む スポンサード リンク
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