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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a set of Johnson circles comprises three
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
s of equal
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
sharing one common point of intersection . In such a configuration the circles usually have a total of four intersections (points where at least two of them meet): the common point 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 osculate, they only have as a common point, and it will then be considered that be their 2-wise intersection as well; if they should coincide we declare their 2-wise intersection be the point diametrically opposite . The three 2-wise intersection points define the reference triangle of the figure. The concept is named after Roger Arthur Johnson.Roger Arthur Johnson (1890–1954)


Properties

# The centers of the Johnson circles lie on a circle of the same radius as the Johnson circles centered at . These centers form the Johnson triangle. # The circle centered at with radius , known as the anticomplementary circle is tangent to each of the Johnson circles. The three tangent points are reflections of point 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 In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is no ...
of the reference triangle. It is similar to the Johnson triangle, and is homothetic by a factor 2 centered at , their common circumcenter. # Johnson's theorem: The 2-wise intersection points of the Johnson circles (vertices of the reference triangle ) lie on a circle of the same radius as the Johnson circles. This property is also well known in
Romania Romania ( ; ro, România ) is a country located at the crossroads of Central, Eastern, and Southeastern Europe. It borders Bulgaria to the south, Ukraine to the north, Hungary to the west, Serbia to the southwest, Moldova to the east, and ...
as The 5 lei coin problem of Gheorghe Țițeica. # The reference triangle is in fact congruent to the Johnson triangle, and is homothetic to it by a factor −1. # The point is the
orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the ' ...
of the reference triangle and the
circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every poly ...
of the Johnson triangle. # The homothetic center of the Johnson triangle and the reference triangle is their common
nine-point center In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circl ...
.


Proofs

Property 1 is obvious from the definition. Property 2 is also clear: for any circle of radius , and any point on it, the circle of radius centered at is tangent to the circle in its point opposite to ; this applies in particular to , giving the anticomplementary circle . Property 3 in the formulation of the homothety immediately follows; the triangle of points of tangency is known as the anticomplementary triangle. For properties 4 and 5, first observe that any two of the three Johnson circles are interchanged by the reflection in the line connecting and their 2-wise intersection (or in their common tangent at if these points should coincide), and this reflection also interchanges the two vertices of the anticomplementary triangle lying on these circles. The 2-wise intersection point therefore is the midpoint of a side of the anticomplementary triangle, and lies on the
perpendicular bisector In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
of this side. Now the midpoints of the sides of any triangle are the images of its vertices by a homothety with factor −½, centered at the barycenter of the triangle. Applied to the anticomplementary triangle, which is itself obtained from the Johnson triangle by a homothety with factor 2, it follows from composition of homotheties that the reference triangle is homothetic to the Johnson triangle by a factor −1. Since such a homothety is a congruence, this gives property 5, and also the Johnson circles theorem since congruent triangles have
circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every pol ...
s of equal radius. For property 6, it was already established that the perpendicular bisectors of the sides of the anticomplementary triangle all pass through the point ; since that side is parallel to a side of the reference triangle, these perpendicular bisectors are also the
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
s of the reference triangle. Property 7 follows immediately from property 6 since the homothetic center whose factor is -1 must lie at the midpoint of the circumcenters  of the reference triangle and  of the Johnson triangle; the latter is the orthocenter of the reference triangle, and its nine-point center is known to be that midpoint. Since the
central symmetry In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
also maps the orthocenter of the reference triangle to that of the Johnson triangle, the homothetic center is also the nine-point center of the Johnson triangle. There is also an algebraic proof of the Johnson circles theorem, using a simple vector computation. There are vectors \vec, \vec, \vec, all of length , such that the Johnson circles are centered respectively at H+\vec, H+\vec, H+\vec. Then the 2-wise intersection points are respectively H+\vec+\vec, H+\vec+\vec, H+\vec+\vec, and the point H+\vec+\vec+\vec clearly has distance to any of those 2-wise intersection points.


Further properties

The three Johnson circles can be considered the reflections of the circumcircle of the reference triangle about each of the three sides of the reference triangle. Furthermore, under the reflections about the three sides of the reference triangle, its orthocenter maps to three points on the circumcircle of the reference triangle that form the vertices of the circum-orthic triangle, its circumcenter maps onto the vertices of the Johnson triangle and its
Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, incl ...
(line passing through ) generates three lines that are concurrent at ''X''(110). The Johnson triangle and its reference triangle share the same nine-point center, the same Euler line and the same
nine-point circle In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of ...
. The six points formed from the vertices of the reference triangle and its Johnson triangle all lie on the Johnson circumconic that is centered at the nine-point center and that has the point ''X''(216) of the reference triangle as its perspector. The circumconic and the circumcircle share a fourth point, ''X''(110) of the reference triangle. Finally there are two interesting and documented circumcubics that pass through the six vertices of the reference triangle and its Johnson triangle as well as the circumcenter, the orthocenter and the nine-point center. The first is known as the first Musselman cubic – ''K''026. This cubic also passes through the six vertices of the
medial triangle In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is no ...
and the medial triangle of the Johnson triangle. The second cubic is known as the Euler central cubic – ''K''044. This cubic also passes through the six vertices of the orthic triangle and the orthic triangle of the Johnson triangle. The ''X''(''i'') point notation is the Clark Kimberling
ETC * Etc. or et cetera, a Latin expression meaning "and the other things" or "and the rest". ETC or etc may also refer to: Companies and organizations * ETC (Chilean TV channel), a Chilean cable television channel * ETC (Philippine TV channel), a ...
classification of triangle centers.


External links

* * F. M. Jackson and * F. M. Jackson and * * * * Bernard Giber
Circumcubic K026
* Bernard Giber

* Clark Kimberling,

. ''(Lists some 3000 interesting points associated with any triangle.)''


References

{{reflist Triangle geometry Circles