Steiner inellipse

;Definition An ellipse that is tangent to the sides of a triangle at its midpoints M_1,M_2,M_3 is called the Steiner inellipse of . Properties: For an arbitrary triangle with midpoints M_1,M_2,M_3 of its sides the following statements are true: a) There exists exactly one Steiner inellipse. b) The center of the Steiner inellipse is the centroid of . c1) The triangle \triangle M_1M_2M_3 has the same centroid and the Steiner inellipse of is the Steiner ellipse of the triangle \triangle M_1M_2M_3. c2) The Steiner inellipse of a triangle is the scaled Steiner Ellipse with scaling factor 1/2 and the centroid as center. Hence both ellipses have the same eccentricity, are similar. d) The area of the Steiner inellipse is \tfrac{\pi}{3 \sqrt{3}}-times the area of the triangle. e) The Steiner inellipse has the greatest area of all inellipses of the triangle. ;Proof 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. 2) Midpoints of sides are mapped onto midpoints and centroids on centroids. The center of an ellipse is mapped onto the center of its image. Hence its suffice to prove properties a),b),c) for an equilateral triangle: a) To any equilateral triangle there exists an incircle. It touches the sides at its midpoints. There is no other (non-degenerate) conic section with the same properties, because a conic section is determined by 5 points/tangents. b) By a simple calculation. c) The circumcircle is mapped by a scaling, with factor 1/2 and the centroid as center, onto the incircle. The eccentricity is an invariant. d) The ratio of areas is invariant to affine transformations. So the ratio can be calculated for the equilateral triangle. e) See Inellipse. == Parametric representation and semi-axes ==

Parametric representation: • Because a Steiner inellipse of a triangle is a scaled Steiner ellipse (factor 1/2, center is centroid) one gets a parametric representation derived from the trigonometric representation of the Steiner ellipse : ::\vec x =\vec p(t)=\overrightarrow{OS}\; +\; {\color{blue}\frac 1 2 }\overrightarrow{SC}\; \cos t \;+\; \frac{1}{{\color{blue}2}\sqrt{3}}\overrightarrow{AB}\; \sin t \; , \quad 0\le t • The 4 vertices of the Steiner inellipse are ::\vec p(t_0),\; \vec p(t_0\pm\frac{\pi}{2}),\; \vec p(t_0+\pi), :where is the solution of ::\cot (2t_0)= \tfrac{\vec f_1^{\, 2}-\vec f_2^{\, 2}}{2\vec f_1 \cdot \vec f_2}\quad with \quad \vec f_1=\frac 1 2 \vec{SC},\quad \vec f_2=\frac{1}{2\sqrt{3}}\vec{AB}\ . Semi-axes: • With the abbreviations ::\begin{align} M &:= {\color{blue} \frac 1 4} \left(\vec{SC}^2+\frac{1}{3}\vec{AB}^2 \right) \\ N &:= \frac{1}{{\color{blue}4}\sqrt{3}} \left|\det \left(\vec{SC},\vec{AB} \right)\right| \end{align} :one gets for the semi-axes (where ): ::\begin{align} a &= \frac{1}{2} \left(\sqrt{M+2N}+\sqrt{M-2N} \right) \\ b &= \frac{1}{2} \left(\sqrt{M+2N}-\sqrt{M-2N} \right)\ . \end{align} • The linear eccentricity of the Steiner inellipse is ::c=\sqrt{a^2-b^2}=\dotsb=\sqrt{\sqrt{M^2-4N^2}}\ . ==Trilinear equation==

The equation of the Steiner inellipse in trilinear coordinates for a triangle with side lengths (with these parameters having a different meaning than previously) is :a^2x^2+b^2y^2+c^2z^2-2abxy-2bcyz-2cazx = 0 where is an arbitrary positive constant times the distance of a point from the side of length , and similarly for and with the same multiplicative constant. ==Other properties==

The lengths of the semi-major and semi-minor axes for a triangle with sides are Denote the centroid and the first and second Fermat points of a triangle as respectively. The major axis of the triangle's Steiner inellipse is the inner bisector of The lengths of the axes are |GF_-| \pm |GF_+|\! ; that is, the sum and difference of the distances of the Fermat points from the centroid. 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. :\frac{\overline{PA} \cdot \overline{QA}}{\overline{CA} \cdot \overline{AB}} + \frac{\overline{PB} \cdot \overline{QB}}{\overline{AB} \cdot \overline{BC}} + \frac{\overline{PC} \cdot \overline{QC}}{\overline{BC} \cdot \overline{CA}} = 1. == Graphical construction ==

The Steiner inellipse of a triangle ABC can be determined using a graphical construction based on establishing an affine transformation between the given triangle and an equilateral triangle. Since this projection preserves the midpoints of the sides, it allows for a graphical correspondence between the incircle of the equilateral triangle and the desired inellipse. The major and minor axes of an ellipse are always perpendicular. The construction uses a circle passing through A and A' (the original and transformed vertices) to find specific points k and k on the axis of affinity. The rays kA and kA identify the unique directions that remain perpendicular after the transformation, which must be the directions of the ellipse's semi-axes. To perform the construction, one side of the triangle is taken as the axis of affinity. (If the triangle is equilateral, the problem is trivial as the inellipse is its incircle; if it is isosceles, one of the two equal sides is typically chosen). The procedure is as follows: • Given triangle ABC, construct the equilateral triangle ''A'BC on side BC''. • Draw the segment ''A'A (the direction of affinity with axis BC'') between the two triangles. • From the midpoint m of ''A'A, draw the perpendicular bisector until it intersects the line BC (or its extension) at point O''. • Draw the circle centered at O that passes through A' and A, which intersects the line BC at points k and k. • The direction of the ray kA coincides with the major axis of the Steiner inellipse, and the ray kA with the minor axis. • The endpoints of the semi-axes are obtained by projecting the points of the incircle of the equilateral triangle—specifically those where the tangents are parallel to rays ''kA' and kA' ''—it relies on the fact that affine transformations preserve parallelism between two lines in the same plane. This same construction is applicable to the Steiner circumellipse, but uses the circumcircle of the equilateral triangle instead of the incircle. ==Generalization==

The Steiner inellipse of a triangle can be generalized to -gons: some -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 -gon. ==References==