Steiner ellipse

In geometry, the Steiner ellipse of a triangle is the unique circumellipse (an ellipse that touches the triangle at its vertices) whose center is the triangle's centroid.

Named after Jakob Steiner, it is an example of a circumconic.

and hence is 4 times the area of the Steiner inellipse.

[1] The Steiner ellipse is the scaled Steiner inellipse (factor 2, center is the centroid).

Hence both ellipses are similar (have the same eccentricity).

A) For an equilateral triangle the Steiner ellipse is the circumcircle, which is the only ellipse, that fulfills the preconditions.

A conic is uniquely determined by 5 points.

Hence the circumcircle is the only Steiner ellipse.

B) Because an arbitrary triangle is the affine image of an equilateral triangle, an ellipse is the affine image of the unit circle and the centroid of a triangle is mapped onto the centroid of the image triangle, the property (a unique circumellipse with the centroid as center) is true for any triangle.

The area of the circumcircle of an equilateral triangle is

An affine map preserves the ratio of areas.

Hence the statement on the ratio is true for any triangle and its Steiner ellipse.

An ellipse can be drawn (by computer or by hand), if besides the center at least two conjugate points on conjugate diameters are known.

The shear mapping with axis

is a vertex of the Steiner ellipse of triangle

This vertex can be determined from the data (ellipse with center

It turns out that Or by drawing: Using de la Hire's method (see center diagram) vertex

of the Steiner ellipse of the isosceles triangle

is fixed, because it is a point on the shear axis.

With help of this pair of conjugate semi diameters the ellipse can be drawn, by hand or by computer.

Wanted: Parametric representation and equation of its Steiner ellipse The centroid of the triangle is

Parametric representation: From the investigation of the previous section one gets the following parametric representation of the Steiner ellipse: The roles of the points for determining the parametric representation can be changed.

Equation: If the origin is the centroid of the triangle (center of the Steiner ellipse) the equation corresponding to the parametric representation

one gets the equation of the Steiner ellipse: If the vertices are already known (see above), the semi axes can be determined.

If one is interested in the axes and eccentricity only, the following method is more appropriate: Let be

the semi axes of the Steiner ellipse.

From Apollonios theorem on properties of conjugate semi diameters of ellipses one gets: Denoting the right hand sides of the equations by

The linear eccentricity of the Steiner ellipse is and the area

in this section with other meanings in this article !

The equation of the Steiner circumellipse in trilinear coordinates is[1] for side lengths a, b, c. The semi-major and semi-minor axes (of a triangle with sides of length a, b, c) have lengths[1] and focal length where The foci are called the Bickart points of the triangle.

The Steiner ellipse of an isosceles triangle . The three line segments inside the triangle are the triangle's medians , each bisecting a side. The medians coincide at the triangle's centroid , which is also the center of the Steiner ellipse.
Steiner ellipse of an equilateral (left) and isosceles triangle
Steps for determining congugate points on a Steiner ellipse:
1) transformation of the triangle onto an isosceles triangle
2) determination of point which is conjugate to (steps 1–5)
3) drawing the ellipse with conjugate half diameters
Steiner ellipse of a triangle including the axes and verices (purple)
Steiner ellipse as example for "equation"