Descartes' fly

Regarding the Steiner ellipses, seen last week, Salva Fuster comments: “The triangle whose vertices are the midpoints of the segments of the original triangle is similar to the original, the ratio of similarity between both being 1/2 (1 /4 if we refer to the area). Since the centroid of both triangles is the same and the medians of both triangles coincide on the same lines, the proportion of areas between the two ellipses will be 1/4, although it seems to me that it would be necessary to demonstrate the uniqueness of the inellipse and also that, constructed as similar to the circumellipse, it is tangent to the midpoints of the segments of the original triangle.”

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Combinatorial designs

This is easier to verify in the particular case that the triangle is equilateral, comparing the axes of both ellipses and seeing that those of the circumellipse are twice those of the inelipse (whose area is π/3√3 for a triangle of area 1).

As for the fraction 13/42, which, multiplied by 15!, gives the total number of solutions to the schoolgirls' problem, it is the result of the following sum:

1/168 + 1/168 + 1/24 + 1/24 + 1/12 + 1/12 + 1/21 = 13/42 (why?).

A genius catching flies

As we saw last week, Jakob Steiner detested analytical geometry, which he considered “impure,” a notable example of how mathematicians, often considered the paradigm of logical thinking, can fall into genuine emotional delusions. Well, analytical geometry, with its fusion of algebra and “pure” geometry, is one of the most powerful mathematical tools.

The antecedents go back to ancient Greece, since both Menaechmus, a disciple of Plato, and Apollonius of Perga, the Great Geometer, used mixed methods very close to analytical geometry as we understand it today; although the clearest precursor was the distinguished Persian poet and mathematician Omar Jayam, whose

Treatise on Demonstrations of Problems in Algebra

, written in the 11th century, can be considered the founding text of analytical geometry. But the one who gave it definitive form was René Descartes, with the help of a fly.

Due to his poor health, Descartes spent a lot of time lying down, and not only did his well-known philosophical reflections emerge from his prostration, but also some important contributions to mathematics. It is said that one day he was lying in bed with his eyes lost when he noticed a fly that was fluttering around the room, and instead of limiting himself to "catching flies", as someone else would have done, he thought that to determine the position of the insect it was enough to with knowing its distance from the ground and from two walls perpendicular to each other (and from the ground, naturally). And if instead of fluttering around the room, the fly had walked on Descartes's presumably rectangular bedside table, it could have determined its position by its distance from two perpendicular sides. Cartesian coordinates were born.

The power of this simple idea lies in the fact that, taking a pair of perpendicular lines as a reference system, we can convert a line into an equation and vice versa. For example, the line in the figure passes through the point of intersection of the axes and any of its points is twice as far from the horizontal axis as it is from the vertical axis. Let us call x (abscissa) the second distance and y (ordinate) the first, and all points on the line will satisfy the relationship y = 2x. Or what is the same: the line is the graphic representation of the equation y = 2x.

Without searching the Internet or dusting off your old school books, can you find the equation of a circle whose center coincides with the point of intersection of the axes and whose radius measures 5 units?

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