Steiner's ellipse | The sport of science | EUROtoday

As we noticed final week, the “schoolgirl problem” admits 7 non-isomorphic options (that’s, with a special construction), listed in 1922 by the American mathematician Frank Nelson Cole (1861-1926), who turned well-known within the early Twenties. twentieth century for locating the elements of the 67th Mersenne quantity (2⁶⁷– 1). Édouard Lucas had proven that M₆₇ was not prime, however he had not been in a position to issue it. And Cole completed the feat of discovering these elements when paper and pencil had been the one calculator obtainable (devoting himself to the issue, as he confessed, each Sunday for 3 years):

M⁶⁷ = 147.573.952.589.676.412.927 = 193.707.721 × 761.838.257.287

And Cole additionally calculated (a trifle in comparison with the earlier calculation) the entire variety of options – together with isomorphic ones – to the schoolgirl downside:

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

Circunelipse and inellipse

In addition to his essential contributions to the idea of combinatorial designs, as we noticed final week, the Swiss mathematician Jakob Steiner (whose “minimal trees” – the bonsai of graphs – we handled 5 years in the past) was one of many best geometers. of all occasions; the best after Apollonius of Perga, in response to some. He detested analytical geometry, which in response to him contaminated “pure” geometry, and his work was based mostly completely on the strategies of artificial and projective geometry, to the event of which he contributed notably.

The reference to Apollonius when speaking about Steiner is particularly pertinent, since, just like the Great Geometer, he made essential contributions to the research of conics. In this area, Steiner is greatest identified for his ellipses circumscribed and inscribed in a triangle.

The Steiner circumellipse is the one ellipse that passes via the three vertices of a triangle and whose middle is the centroid or centroid of the identical (keep in mind that the centroid of a triangle is the purpose of intersection of its medians, which coincides with its middle of gravity if we think about it a bodily object).

Someone might imagine {that a} circle can be an ellipse and that, due to this fact, the circumcircle of a triangle would even be a Steiner circumellipse. But this isn’t the case, because the middle of the circumscribed circle (circumcenter) is the purpose of intersection of the bisectors of the triangle, not its medians (the reason being apparent: all of the factors of the bisector on all sides are equidistant from the 2 vertices). equivalent to that facet, so the purpose of intersection of the bisectors is equidistant from the three vertices).

Among different properties, the Steiner circumellipse is, of all of the ellipses circumscribed by a triangle, the one with the smallest space (are you able to calculate it based mostly on the world of ​​the triangle?).

When we discuss Steiner's ellipse with out specifying the rest, we’re referring to its circumellipse, which shouldn’t be confused with the inellipse. The Steiner inellipse is the ellipse inscribed in a triangle that’s tangent to the midpoints of its sides (and it’s justified to say “the” as a result of it’s distinctive). The floor space of ​​the Steiner inellipse is 1 / 4 of that of the Steiner circumellipse (are you able to show it?).

I recommend to my sagacious readers that they start by analyzing the actual and far less complicated case of the circumellipse and the inellipse of an equilateral triangle.

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