In
triangle geometry
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collin ...
, the Kiepert conics are two special
conic
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
s associated with the reference triangle. One of them is a
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
, called the Kiepert hyperbola and the other is a
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
, called the Kiepert parabola. The Kiepert conics are defined as follows:
:If the three triangles
,
and
, constructed on the sides of a triangle
as bases, are similar, isosceles and similarly situated, then the triangles
and
are in
perspective. As the base angle of the isosceles triangles varies between
and
, the
locus
Locus (plural loci) is Latin for "place". It may refer to:
Entertainment
* Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front
* ''Locus'' (magazine), science fiction and fantasy magazine
** ''Locus Award' ...
of the
center of perspectivity of the triangles
and
is a hyperbola called the Kiepert hyperbola and the
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
of their
axis of perspectivity is a parabola called the Kiepert parabola.
It has been proved that the Kiepert hyperbola is the hyperbola passing through the vertices, the
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
and 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 Kiepert parabola is the parabola
inscribed
{{unreferenced, date=August 2012
An inscribed triangle of a circle
In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...
in the reference triangle having the
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, includ ...
as
directrix and the triangle center X
110 as
focus
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
. The following quote from a paper by R. H. Eddy and R. Fritsch is enough testimony to establish the importance of the Kiepert conics in the study of triangle geometry:
:"If a visitor from Mars desired to learn the geometry of the triangle but could stay in the earth's relatively dense atmosphere only long enough for a single lesson, earthling mathematicians would, no doubt, be hard-pressed to meet this request. In this paper, we believe that we have an optimum solution to the problem. The Kiepert conics ...."
Kiepert hyperbola
The Kiepert hyperbola was discovered by
Ludvig Kiepert while investigating the solution of the following problem proposed by
Emile Lemoine
Emil or Emile may refer to:
Literature
*''Emile, or On Education'' (1762), a treatise on education by Jean-Jacques Rousseau
* ''Émile'' (novel) (1827), an autobiographical novel based on Émile de Girardin's early life
*''Emil and the Detective ...
in 1868: "Construct a triangle, given the peaks of the equilateral triangles constructed on the sides." A solution to the problem was published by
Ludvig Kiepert in 1869 and the solution contained a remark which effectively stated the locus definition of the Kiepert hyperbola alluded to earlier.
Basic facts
Let
be the side lengths and
the vertex angles of the reference triangle
.
Equation
The equation of the Kiepert hyperbola in barycentric coordinates
is
:
Center, asymptotes
*The centre of the Kiepert hyperbola is the triangle center X(115). The barycentric coordinates of the center are
:
.
*The asymptotes of the Kiepert hyperbola are the
Simson line
In geometry, given a triangle and a point on its circumcircle, the three closest points to on lines , , and are collinear. The line through these points is the Simson line of , named for Robert Simson. The concept was first published, howeve ...
s of the intersections of the
Brocard axis
In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This ...
with the
circumcircle
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 polyg ...
.
*The Kiepert hyperbola is a
rectangular hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
and hence its eccentricity is
.
Properties
#The center of the Kiepert hyperbola lies on the
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 ea ...
. The center is the midpoint of the line segment joining the isogonic centers of triangle
which are the triangle centers X(13) and X(14) in the Encyclopedia of Triangle Centers.
#The image of the Kiepert hyperbola under the
isogonal transformation is the
Brocard axis
In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This ...
of triangle
which is the line joining the
symmedian point
In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the co ...
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 polyg ...
.
#Let
be a point in the plane of a nonequilateral triangle
and let
be the trilinear polar of
with respect to
. The locus of the points
such that
is perpendicular to the Euler line of
is the Kiepert hyperbola.
Kiepert parabola
The Kiepert parabola was first studied in 1888 by a German mathematics teacher Augustus Artzt in a "school program".
Basic facts
*The equation of the Kiepert parabola in barycentric coordinates
is
::
where
.
*The focus of the Kiepert parabola is the triangle center X(110). The barycentric coordinates of the focus are
::
*The directrix of the Kiepert parabola is the Euler line of triangle
.
Images
KiepertHyperbola01.png, Kiepert hyperbola showing the center of perspectivity of triangles ABC and A'B'C'
KiepertHyperbola02.png, Kiepert hyperbola showing the orthocenter, the incenter and the perpendicular asymptotes
KiepertParabola.png, Kiepert parabola of triangle ABC. The figure also shows a member (line LMN) of the family of lines whose envelope is the Kiepert parabola.
KiepertParabola01.png, Kiepert parabola showing the focus and the directrix
See also
*
Triangle conic
*
Modern triangle geometry
In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a triangle discovered and developed roughly since the beginning of the last quarter of the nineteenth century. Triangles and ...
External links
*
*
References
{{reflist
Triangle geometry