MaximalTrianglesProperties

Maximal triangles properties

Given the ellipse (e) with axes a , b (a >b, equation x²/a² + y²/b² =1, c^2 = a^2-b^2). Consider the maximal triangles ABC inscribed in it. Their circumcenters O lie on an ellipse (f), similar to (e), with the same center as (e) and axes in the same directions as (e) but reversed in magnitude. The great axis of (f): a1 = c^2/(4*b) along the small axis of (e), and small axis b2 = c^2/(4*a). Their orthocenters H lie on an ellipse (g) homothetic to (f) by the factor 2. The heights of ABC coincide with the normals of (e) at the vertices of the triangle. (Steiner Werke Bd. II, p. 347). The diameter PQ of the circumcircle (h) of ABC, passing through the center M of (e) is divided by it in two parts PM, MQ whose product is PM*MQ = (a^2+b^2)/2.

The basic properties of the maximal triangles inscribed in (e) are discussed in MaximalTrianglesInEllipse.html . The statement on the ellipse (f) follows from an easy calculation. That on (g) follows from the well known relation MH = 2*OM on the Euler line OH. M is the centroid of all maximal triangles. The radius r of the circle (h) can be calculated and the product PM*MQ = (r+OM)*(r-OM) gives the formula after a calculation. Steiner goes further in the cited reference and gives the maximum and minimum values of the radius r=r(A) and the corresponding product p = p(A)=AB*BC*CA of sides of the triangle:

Look at the file CircumcentersLocus.html for the story of the shape and a parametrization of the locus of circumcenters O.

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