Steiner inellipse

By comparison the inscribed circle and Mandart inellipse of a triangle are other inconics that are tangent to the sides, but not at the midpoints unless the triangle is equilateral.

[3] The Steiner inellipse contrasts with the Steiner circumellipse, also called simply the Steiner ellipse, which is the unique ellipse that passes through the vertices of a given triangle and whose center is the triangle's centroid.

[4] An ellipse that is tangent to the sides of a triangle △ABC at its midpoints

of its sides the following statements are true: a) There exists exactly one Steiner inellipse.

[5]: p.146 [6]: Corollary 4.2 The proofs of properties a),b),c) are based on the following properties of an affine mapping: 1) any triangle can be considered as an affine image of an equilateral triangle.

c) The circumcircle is mapped by a scaling, with factor 1/2 and the centroid as center, onto the incircle.

d) The ratio of areas is invariant to affine transformations.

Parametric representation: Semi-axes: The equation of the Steiner inellipse in trilinear coordinates for a triangle with side lengths a, b, c (with these parameters having a different meaning than previously) is[1] where x is an arbitrary positive constant times the distance of a point from the side of length a, and similarly for b and c with the same multiplicative constant.

The lengths of the semi-major and semi-minor axes for a triangle with sides a, b, c are[1] where According to Marden's theorem,[3] if the three vertices of the triangle are the complex zeros of a cubic polynomial, then the foci of the Steiner inellipse are the zeros of the derivative of the polynomial.

The major axis of the Steiner inellipse is the line of best orthogonal fit for the vertices.

[6]: Corollary 2.4 Denote the centroid and the first and second Fermat points of a triangle as ⁠

The major axis of the triangle's Steiner inellipse is the inner bisector of ⁠

that is, the sum and difference of the distances of the Fermat points from the centroid.[7]: Thm.

1 The axes of the Steiner inellipse of a triangle are tangent to its Kiepert parabola, the unique parabola that is tangent to the sides of the triangle and has the Euler line as its directrix.[7]: Thm.

3 The foci of the Steiner inellipse of a triangle are the intersections of the inellipse's major axis and the circle with center on the minor axis and going through the Fermat points.[7]: Thm.

6 As with any ellipse inscribed in a triangle △ABC, letting the foci be P and Q we have[8] The Steiner inellipse of a triangle can be generalized to n-gons: some n-gons have an interior ellipse that is tangent to each side at the side's midpoint.

Marden's theorem still applies: the foci of the Steiner inellipse are zeroes of the derivative of the polynomial whose zeroes are the vertices of the n-gon.