The Steiner ellipse

As we saw last week, the “schoolgirl problem” admits 7 non-isomorphic solutions (that is, with a different structure), listed in 1922 by the American mathematician Frank Nelson Cole (1861-1926), who became famous in the early 1920s. 20th century for finding the factors of the 67th Mersenne number (2⁶⁷– 1). Édouard Lucas had shown that M₆₇ was not prime, but he had not been able to factor it. And Cole accomplished the feat of finding those factors when paper and pencil were the only calculator available (devoting himself to the problem, as he confessed, every Sunday for three years):

M⁶⁷ = 147,573,952,589,676,412,927 = 193,707,721 × 761,838,257,287

And Cole also calculated (a trifle compared to the previous calculation) the total number of solutions – including isomorphic ones – to the schoolgirl problem:

fifteen! x 13/42 = 404,756,352,000 (how do you get this number?).

Circunelipse and inelipse

In addition to his important contributions to the theory of combinatorial designs, as we saw last week, the Swiss mathematician Jakob Steiner (whose “minimal trees” – the bonsai of graphs – we discussed five years ago) was one of the greatest geometers. of all times; the greatest after Apollonius of Perga, according to some. He detested analytical geometry, which according to him contaminated “pure” geometry, and his work was based exclusively on the methods of synthetic and projective geometry, to the development of which he contributed notably.

The reference to Apollonius when talking about Steiner is especially pertinent, since, like the Great Geometer, he made important contributions to the study of conics. In this field, Steiner is best known for his ellipses circumscribed and inscribed in a triangle.

The Steiner circumellipse is the only ellipse that passes through the three vertices of a triangle and whose center is the centroid or centroid of the same (remember that the centroid of a triangle is the point of intersection of its medians, which coincides with its center of gravity if we consider it a physical object).

Someone may think that a circle is also an ellipse and that, therefore, the circumcircle of a triangle would also be a Steiner circumellipse. But this is not the case, since the center of the circumscribed circle (circumcenter) is the point of intersection of the bisectors of the triangle, not of its medians (the reason is obvious: all the points of the bisector of each side are equidistant from the two vertices). corresponding to that side, so the point of intersection of the bisectors is equidistant from the three vertices).

Among other properties, the Steiner circumellipse is, of all the ellipses circumscribed by a triangle, the one with the smallest area (can you calculate it based on the area of ​​the triangle?).

When we talk about Steiner's ellipse without specifying anything else, we are referring to its circumellipse, which should not be confused with the inellipse. The Steiner inellipse is the ellipse inscribed in a triangle that is tangent to the midpoints of its sides (and it is justified to say “the” because it is unique). The surface area of ​​the Steiner inellipse is a quarter of that of the Steiner circumellipse (can you prove it?).

I suggest to my sagacious readers that they begin by analyzing the particular and much simpler case of the circumellipse and the inellipse of an equilateral triangle.

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