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Concept
Steiner ellipse
Summary
In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse (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. 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.
The area of the Steiner ellipse equals the area of the triangle times and hence is 4 times the area of the Steiner inellipse. The Steiner ellipse has the least area of any ellipse circumscribed about the triangle.
The Steiner ellipse is the scaled Steiner inellipse (factor 2, center is the centroid). Hence both ellipses are similar (have the same eccentricity).
A Steiner ellipse is the only ellipse, whose center is the centroid of a triangle and contains the points . The area of the Steiner ellipse is -fold of the triangle's area.
Proof
A) For an equilateral triangle the Steiner ellipse is the circumcircle, which is the only ellipse, that fulfills the preconditions. The desired ellipse has to contain the triangle reflected at the center of the ellipse. This is true for the circumcircle. 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 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 -fold of the area of the triangle. 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.
Official source
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