MaximalTrianglesInEllipse

Maximal triangles in ellipse

Show that in an ellipse (e) with axes a, b one can inscribe infinitely many triangles A*B*C* of maximal area. All these triangles having area equal to (3*sqrt(3)/4)*a*b. Furthermore, every point A* of the ellipse is a vertex of a maximal triangle. The median from A* is conjugate to the direction of B*C* and the centroid of the triangle coincides with the center of the ellipse. The middles of the sides of the maximal triangles lie on an (1/2)-homothetic ellipse with the same center as (e). [J. Steiner, Lehrsaetze und Aufgaben, Werke Bd. II p. 347]

For a quick proof of the statements consider the affinity f mapping the unit square to the maximal rectangle of the ellipse (see MaximalRectInEllipse.html ). The ellipse is the map under f of the unit circle. The maximal triangles inscribed in the ellipse are the images of the equilaterals inscribed in the unit circle. The statements made are consequences of the conservation, by f, of the parallelity and the ratios along a line. Note that the ellipse (e) is the outer Steiner ellipse of all the maximal triangles and the inner Steiner ellipse of the circumscribed triangle with sides tangent to (e) at the points A*, B* and C*. Since the normals of (e) at A*, B* and C* are the heights of triangle A*B*C* they intersect at a point F, which is also the circumcenter of the tangential triangle.

Some further properties (also due to Steiner) of the maximal triangles are discussed in the file MaximalTrianglesProperties.html .
The maximality property of the equilateral in its circumcircle, together with a related problem is discussed in TriangleSymmetrization.html .
For another point of view of the same subject see the file Steiner_Ellipse.html . Here the starting object is the ellipse. There it is the inscribed maximal triangle.

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