Viviani’s theorem | The science sport | EUROtoday
Regarding Fermat-Torricelli’s level, our common commentator Salva Fuster says:
“A simple construction that allows us to find the Fermat-Torricelli point for the case of a triangle whose largest angle is less than 120º consists of drawing two equilateral triangles (towards the outside) on any pair of sides of the triangle. Subsequently, we join each of the two new vertices with the vertex furthest from the original triangle. The two segments intersect at the Fermat-Torricelli point. And he adds: “I think that a very interesting way to see that the FT point is the one that presents angles of 120º between the point and the vertices of the triangle is the following:
—We draw three semilines that start from the same point and that form 120º between them.
—We couple the vertices of the triangle we want (its largest angle must not exceed 120º) so that they are on those three rays.
—We draw at each vertex, a perpendicular to the ray that contains it, forming an equilateral triangle.
—Taking into account Viviani’s theorem, we can see that any other candidate point of FT that is not the origin of the three ray lines will have a greater total distance.”
Viviani’s theorem, which owes its name to the Italian mathematician and physicist Vincenzo Viviani (1622-1703), says that the sum of the distances from an interior point to each of the sides of an equilateral triangle is equal to the height of the triangle (Can you think of a simple demonstration?).
Viviani’s theorem can be generalized to all equilateral polygons and equiangular polygons: the sum of the distances from any interior point to the sides of an equilateral or equiangular polygon is constant.
Vincenzo Viviani is known, above all, because he was a long-time collaborator and confidant of Galileo, whose first biography he is the author of. He was also the first to determine the speed of sound, and he preceded Foucault by two centuries in constructing the pendulum named after the French physicist.
Viviani’s window
Less known outside the specialized field is the architectural problem known as “Viviani’s window”, which the Florentine mathematician posed on the finish of the seventeenth century, and which was addressed, amongst others, by Leibniz and Bernoulli. It consists of opening 4 equal home windows in a hemispherical dome in order that the remaining floor of the dome is quadrable (a determine is alleged to be “squareable” whether it is attainable to acquire, from it and by geometric strategies, a sq. that has the identical space, as within the well-known—and unimaginable—case of squaring the circle). The answer is the intersection of the dome with a cylinder whose radius is half that of the sphere (however that’s one other article).

Varignon’s artifact
Regarding Torricelli’s point, Susana Luu comments:
“Apart from the Steiner tree, a problem that I did not know about, another obvious generalization of that problem is: given n points, not necessarily three, calculate a point such that the sum of the distances from it to the n points is minimum. I encountered this problem a long time ago, and I really liked one way of calculating the solution: interpreting it as a problem in physics, like the Varignon artifact.”
Can you think of a simple way to turn the determination of the Torricelli point into a physics problem?
By the best way (and as a touch), Pierre Varignon (1654-1722) was a French mathematician and physicist who made vital contributions to statics, and really particularly to equilibrium circumstances in three dimensions. He can be recognized for the theory that bears his identify, which says that the midpoints of the perimeters of any quadrilateral are the vertices of a parallelogram (are you able to show it?).
https://elpais.com/ciencia/el-juego-de-la-ciencia/2025-01-17/el-teorema-de-viviani.html