Regarding Steiner's ellipses, seen last week, Salva-Foster comments: “A triangle whose vertices are the midpoints of the parts of the original triangle is similar to the original triangle, and their similarity ratio is 1/2 (1/4 if we refer to the area). Since The centroid of both triangles is the same and the averages of both triangles coincide on the same lines, the ratio of the areas between the ellipse will be 1/4, although it seems to me that it would be necessary to prove the uniqueness of the two shapes and also the ellipse, which is constructed similarly to the ellipse The circumference is tangent to the midpoint of the original triangle.

It is easier to verify this in the specific case that the triangle is equilateral, by comparing the axes of the ellipse and seeing that the axes of the circumscribed ellipse are twice the axis of the ellipse (which has an area of 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 students’ 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 at catching flies

As we saw last week, Jacob Steiner hated analytic geometry, which he considered “impure,” and is a prime example of how mathematicians, often held up as paragons of logical thinking, fall into genuine emotional delusions. Well, analytical geometry, with its combination of algebra and “pure” geometry, is one of the most powerful mathematical tools.

Precedents go back to ancient Greece, where Menachemus, a student of Plato, and Apollonius of Perga, the great geometer, used mixed methods very close to analytical geometry as we understand it today; Although the most obvious predecessor was the distinguished Persian poet and mathematician Omar Gayam, who *A treatise on proofs of algebra problems*Written in the 11th century, it can be considered the foundational text of analytical geometry. But the one who gave it final form was Rene Descartes, with the help of a fly.

Due to his poor health, Descartes spent a lot of time lying down, and from his prostration not only came his well-known philosophical reflections, but also some important contributions to mathematics. It is said that one day he was lying in bed with his eyes closed when he noticed a fly flitting across the room, and instead of confining himself to “catching flies,” as anyone else would have done, he thought that to determine the position of the insect it was sufficient to know its distance from the floor and from two walls. Perpendicular to each other (and from the ground, of course). If, instead of flying across the room, the fly had walked across Descartes' rectangular bedside table, its position could have been determined by its distance from two perpendicular sides. Cartesian coordinates are generated.

The power of this simple idea lies in the fact that by taking a pair of perpendicular lines as a reference system, we can transform the 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 x (the ordinate) call x the second distance and y (the ordinate) the first distance, and all points on the line will satisfy the relationship y = 2x. Or what is the same: The line is the graph of the equation y = 2x.

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

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