Start with a non equilateral triangle ABC and apply several times the following symmetrization procedure. If two sides BA, BC say, are non-equal replace them with two equal by taking B' to be the intersection of the medial line of AC and the circumcircle. The resulting triangle AB'C has the same circumcircle and greater area. It may happen that after a finite number of steps the triangle becomes equilateral. When is this the case? When the procedure can be continued infinitely? How fast converges then the triangle to an equilateral one?
In the figure below the triangle DEF results from ABC by applying 5 times the procedure.
The problem is related to the one of finding the triangle with maximal area inscribed in a given circle. The answer is the equilateral. Since, applying the previous procedure to one with two non equal sides gives a triangle with greater area. See the file MaximalTrianglesInEllipse.html for an application of this maximality property of equilaterals.
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Triangle Symmetrization Problem ![[0_0]](TriangleSymmetrization_files/TriangleSymmetrization_0_0_0.png)
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