In triangle geometry, an inellipse is an ellipse that touches the three sides of a triangle. The simplest example is the incircle. Further important inellipses are the Steiner inellipse, which touches the triangle at the midpoints of its sides, the Mandart inellipse and Brocard inellipse (see examples section). For any triangle there exist an infinite number of inellipses.

Example of an inellipse

The Steiner inellipse plays a special role: Its area is the greatest of all inellipses.

Because a non-degenerate conic section is uniquely determined by five items out of the sets of vertices and tangents, in a triangle whose three sides are given as tangents one can specify only the points of contact on two sides. The third point of contact is then uniquely determined.

Parametric representations, center, conjugate diametersEdit

 
An inellipse of a triangle is uniquely determined by the vertices of the triangle and two points of contact  .

The inellipse of the triangle with vertices

 

and points of contact

 

on   and   respectively can by described by the rational parametric representation

  •  

where   are uniquely determined by the choice of the points of contact:

 

The third point of contact is

 

The center of the inellipse is

 

The vectors

 
 

are two conjugate half diameters and the inellipse has the more common trigonometric parametric representation

  •  
 
Brianchon point  

The Brianchon point of the inellipse (common point   of the lines  ) is

 

Varying   is an easy option to prescribe the two points of contact  . The given bounds for   guarantee that the points of contact are located on the sides of the triangle. They provide for   the bounds  .

Remark: The parameters   are neither the semiaxes of the inellipse nor the lengths of two sides.

ExamplesEdit

 
Mandart inellipse

Steiner inellipseEdit

For   the points of contact   are the midpoints of the sides and the inellipse is the Steiner inellipse (its center is the triangle's centroid).

IncircleEdit

For   one gets the incircle of the triangle with center

 

Mandart inellipseEdit

For   the inellipse is the Mandart inellipse of the triangle. It touches the sides at the points of contact of the excircles (see diagram).

 
Brocard inellipse

Brocard inellipseEdit

For   one gets the Brocard inellipse. It is uniquely determined by its Brianchon point given in trilinear coordinates  .

Derivations of the statementsEdit

 
Determination of the inellipse by solving the problem for a hyperbola in an  - -plane and an additional transformation of the solution into the x-y-plane.   is the center of the sought inellipse and   two conjugate diameters. In both planes the essential points are assigned by the same symbols.   is the line at infinity of the x-y-plane.
New coordinates

For the proof of the statements one considers the task projectively and introduces convenient new inhomogene  - -coordinates such that the wanted conic section appears as a hyperbola and the points   become the points at infinity of the new coordinate axes. The points   will be described in the new coordinate system by   and the corresponding line has the equation  . (Below it will turn out, that   have indeed the same meaning introduced in the statement above.) Now a hyperbola with the coordinate axes as asymptotes is sought, which touches the line  . This is an easy task. By a simple calculation one gets the hyperbola with the equation  . It touches the line   at point  .

Coordinate transformation

The transformation of the solution into the x-y-plane will be done using homogeneous coordinates and the matrix

 .

A point   is mapped onto

 

A point   of the  - -plane is represented by the column vector   (see homogeneous coordinates). A point at infinity is represented by  .

Coordinate transformation of essential points
 
 
(One should consider:  ; see above.)

  is the equation of the line at infinity of the x-y-plane; its point at infinity is  .

 

Hence the point at infinity of   (in  - -plane) is mapped onto a point at infinity of the x-y-plane. That means: The two tangents of the hyperbola, which are parallel to  , are parallel in the x-y-plane, too. Their points of contact are

 

Because the ellipse tangents at points   are parallel, the chord   is a diameter and its midpoint the center   of the ellipse

 

One easily checks, that   has the  - -coordinates

 

In order to determine the diameter of the ellipse, which is conjugate to  , in the  - -plane one has to determine the common points   of the hyperbola with the line through   parallel to the tangents (its equation is  ). One gets  . And in x-y-coordinates:

 

From the two conjugate diameters   there can be retrieved the two vectorial conjugate half diameters

 

and at least the trigonometric parametric representation of the inellipse:

 

Analogously to the case of a Steiner ellipse one can determine semiaxes, eccentricity, vertices, an equation in x-y-coordinates and the area of the inellipse.

The third touching point   on   is:

 

The Brianchon point of the inellipse is the common point   of the three lines  . In the  - -plane these lines have the equations:  . Hence point   has the coordinates:

 

Transforming the hyperbola   yields the rational parametric representation of the inellipse:

 
Incircle
 
Incircle of a triangle

For the incircle there is  , which is equivalent to

(1)  Additionally
(2) . (see diagram)

Solving these two equations for   one gets

(3) 

In order to get the coordinates of the center one firstly calculates using (1) und (3)

 

Hence

 
Mandart inellipse

The parameters   for the Mandart inellipse can be retrieved from the properties of the points of contact (see de: Ankreis).

Brocard inellipse

The Brocard inellipse of a triangle is uniquely determined by its Brianchon point given in trilinear coordinates  .[1] Changing the trilinear coordinates into the more convenient representation   (see trilinear coordinates) yields  . On the other hand, if the parameters   of an inellipse are given, one calculates from the formula above for  :  . Equalizing both expressions for   and solving for   yields

 

Inellipse with the greatest areaEdit

  • The Steiner inellipse has the greatest area of all inellipses of a triangle.
Proof

From Apollonios theorem on properties of conjugate semi diameters   of an ellipse one gets:

  (see article on Steiner ellipse).

For the inellipse with parameters   one gets

 
 

where  .
In order to omit the roots, it is enough to investigate the extrema of function  :

 

Because   one gets from the exchange of s and t:

 

Solving both equations for s and t yields

  which are the parameters of the Steiner inellipse.
 
Three mutually touching inellipses of a triangle

See alsoEdit

ReferencesEdit

  1. ^ Imre Juhász: Control point based representation of inellipses of triangles, Annales Mathematicae et Informaticae 40 (2012) pp. 37–46, p.44

External linksEdit