Inellipse

For any triangle there exist an infinite number of inellipses.

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.

The inellipse of the triangle with vertices and points of contact on

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 The Brianchon point of the inellipse (common point

is an easy option to prescribe the two points of contact

guarantee that the points of contact are located on the sides of the triangle.

are neither the semiaxes of the inellipse nor the lengths of two sides.

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

It touches the sides at the points of contact of the excircles (see diagram).

It is uniquely determined by its Brianchon point given in trilinear 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.

Now a hyperbola with the coordinate axes as asymptotes is sought, which touches the line

By a simple calculation one gets the hyperbola with the equation

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

-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

-coordinates In order to determine the diameter of the ellipse, which is conjugate to

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.

yields the rational parametric representation of the inellipse: For the incircle there is

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

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

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

yields From Apollonios theorem on properties of conjugate semi diameters

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

one gets from the exchange of s and t: Solving both equations for s and t yields