Steiner inellipse

Last updated October 29, 2024

In geometry, the Steiner inellipse,^[1]midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. It is an example of an 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. The Steiner inellipse is attributed by Dörrie^[2] to Jakob Steiner, and a proof of its uniqueness is given by Dan Kalman.^[3]

Definition and properties

Definition

An ellipse that is tangent to the sides of a triangle $△ ABC$ at its midpoints $M_{1},M_{2},M_{3}$ is called the Steiner inellipse of $△ ABC$ .

.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}
Arbitrary triangle ^ABC
Steiner inellipse
Steiner ellipse
Major and minor axes Steiner-inellipse-1.svg — Arbitrary triangle $△ ABC$
  Steiner inellipse
  Steiner ellipse
  Major and minor axes

Equilateral triangle ^ABC
Steiner inellipse
Steiner ellipse Steiner-inellipse-0.svg — Equilateral triangle $△ ABC$
  Steiner inellipse
  Steiner ellipse

Properties:
For an arbitrary triangle $△ ABC$ 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 $S$ of $△ ABC$ .
c1) The triangle $\triangle M_{1}M_{2}M_{3}$ has the same centroid $S$ and the Steiner inellipse of $△ ABC$ is the Steiner ellipse of the triangle $\triangle M_{1}M_{2}M_{3}.$
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. ^[5]^: p.146^[6]^{: Corollary 4.2}

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 $△ ABC$ 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\leq t<2\pi \ .

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

t 0

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{aligned}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{aligned}}

one gets for the semi-axes

a, b

(where

a > b

):

{\begin{aligned}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{aligned}}

The linear eccentricity $c$ 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 $a, b, c$ (with these parameters having a different meaning than previously) is^[1]

a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2}-2abxy-2bcyz-2cazx=0

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.

Other properties

The lengths of the semi-major and semi-minor axes for a triangle with sides $a, b, c$ are^[1]

{\frac {1}{6}}{\sqrt {a^{2}+b^{2}+c^{2}\pm 2Z}},

where

Z={\sqrt {a^{4}+b^{4}+c^{4}-a^{2}b^{2}-b^{2}c^{2}-c^{2}a^{2}}}.

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 ⁠ $G,F_{+},F_{-}$ ⁠ respectively. The major axis of the triangle's Steiner inellipse is the inner bisector of ⁠ $\angle F_{+}GF_{-}.$ ⁠ 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.^[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]

{\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.

Generalization

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.^[9]

Related Research Articles

<span class="mw-page-title-main">Ellipse</span> Plane curve: conic section

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity $, a number ranging from to .$

In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle. The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex; the shortest segment between the base and apex is the height. The area of a triangle equals one-half the product of height and base length.

<span class="mw-page-title-main">Bisection</span> Division of something into two equal or congruent parts

In geometry, bisection is the division of something into two equal or congruent parts. Usually it involves a bisecting line, also called a bisector. The most often considered types of bisectors are the segment bisector, a line that passes through the midpoint of a given segment, and the angle bisector, a line that passes through the apex of an angle . In three-dimensional space, bisection is usually done by a bisecting plane, also called the bisector.

In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.

<span class="mw-page-title-main">Centroid</span> Mean position of all the points in a shape

In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any object in $-dimensional Euclidean space.$

<span class="mw-page-title-main">Euler line</span> Line constructed from a triangle

In geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.

<span class="mw-page-title-main">Incenter</span> Center of the inscribed circle of a triangle

In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.

<span class="mw-page-title-main">Midpoint</span> Point on a line segment which is equidistant from both endpoints

In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

<span class="mw-page-title-main">Median (geometry)</span> Line segment joining a triangles vertex to the midpoint of the opposite side

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex, and they all intersect at the triangle's centroid. In the case of isosceles and equilateral triangles, a median bisects any angle at a vertex whose two adjacent sides are equal in length. The concept of a median extends to tetrahedra.

In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name.

In Euclidean geometry, a tangential quadrilateral or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons.

In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial $P$ and the roots of its derivative $P'$ . The set of roots of a real or complex polynomial is a set of points in the complex plane. The theorem states that the roots of $P'$ all lie within the convex hull of the roots of $P$ , that is the smallest convex polygon containing the roots of $P$ . When $P$ has a single root then this convex hull is a single point and when the roots lie on a line then the convex hull is a segment of this line. The Gauss–Lucas theorem, named after Carl Friedrich Gauss and Félix Lucas, is similar in spirit to Rolle's theorem.

In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.

In geometry, the Mandart inellipse of a triangle is an ellipse that is inscribed within the triangle, tangent to its sides at the contact points of its excircles. The Mandart inellipse is named after H. Mandart, who studied it in two papers published in the late 19th century.

In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative. See also geometrical properties of polynomial roots.

In geometry, the Steiner ellipse of a triangle is the unique circumellipse whose center is the triangle's centroid. It is also called the Steiner circumellipse, to distinguish it from the Steiner inellipse. Named after Jakob Steiner, it is an example of a circumconic. By comparison the circumcircle of a triangle is another circumconic that touches the triangle at its vertices, but is not centered at the triangle's centroid unless the triangle is equilateral.

<span class="mw-page-title-main">Inellipse</span> Ellipse tangent to all sides of a triangle

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. For any triangle there exist an infinite number of inellipses.

References

1 2 3 Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html.
↑ H. Dörrie, 100 Great Problems of Elementary Mathematics, Their History and Solution (trans. D. Antin), Dover, New York, 1965, problem 98.
1 2 Kalman, Dan (2008), "An elementary proof of Marden's theorem" (PDF), American Mathematical Monthly , 115 (4): 330–338, doi:10.1080/00029890.2008.11920532, JSTOR 27642475, MR 2398412, archived from the original (PDF) on 2012-08-26.
↑ Weisstein, Eric W. "Steiner Circumellipse". MathWorld .
↑ Chakerian, G. D. (1979), "A distorted view of geometry", in Honsberger, Ross (ed.), Mathematical plums, The Dolciani Mathematical Expositions, vol. 4, Washington, D.C.: Mathematical Association of America, pp. 135–136, 145–146.
1 2 Minda, D.; Phelps, S. (2008), "Triangles, ellipses, and cubic polynomials" (PDF), American Mathematical Monthly , 115 (8): 679–689, doi:10.1080/00029890.2008.11920581, MR 2456092 .
1 2 3 Scimemi, Benedetto, "Simple Relations Regarding the Steiner Inellipse of a Triangle", Forum Geometricorum 10, 2010: 55–77.
↑ Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity", Mathematical Gazette 96, March 2012, 161-165.
↑ Parish, James L., "On the derivative of a vertex polynomial", Forum Geometricorum 6, 2006, pp. 285–288: Proposition 5.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Weisstein-1] 1 2 3 Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html.

[2] H. Dörrie, 100 Great Problems of Elementary Mathematics, Their History and Solution (trans. D. Antin), Dover, New York, 1965, problem 98.

[Kalman-3] 1 2 Kalman, Dan (2008), "An elementary proof of Marden's theorem" (PDF), American Mathematical Monthly , 115 (4): 330–338, doi:10.1080/00029890.2008.11920532, JSTOR 27642475, MR 2398412, archived from the original (PDF) on 2012-08-26.

[4] Weisstein, Eric W. "Steiner Circumellipse". MathWorld .

[Chakerian-5] Chakerian, G. D. (1979), "A distorted view of geometry", in Honsberger, Ross (ed.), Mathematical plums, The Dolciani Mathematical Expositions, vol. 4, Washington, D.C.: Mathematical Association of America, pp. 135–136, 145–146.

[Minda-6] 1 2 Minda, D.; Phelps, S. (2008), "Triangles, ellipses, and cubic polynomials" (PDF), American Mathematical Monthly , 115 (8): 679–689, doi:10.1080/00029890.2008.11920581, MR 2456092 .

[Scimemi-7] 1 2 3 Scimemi, Benedetto, "Simple Relations Regarding the Steiner Inellipse of a Triangle", Forum Geometricorum 10, 2010: 55–77.

[8] Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity", Mathematical Gazette 96, March 2012, 161-165.

[9] Parish, James L., "On the derivative of a vertex polynomial", Forum Geometricorum 6, 2006, pp. 285–288: Proposition 5.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]