The circumference of the nine points | THE GAME OF SCIENCE

In the last two weeks we have talked about the four official “notable points” of the triangle: Incenter, Circumcentro, Ortocenter and Baricentro; But, obviously, there are other singular points, such as the midpoints of the sides (m, n, p). And the intersection points of the three heights with their respective bases (E, G, J) are also special. And what makes them double special is that there is a circumference that passes through all of them. And, even more difficult, this circumference also passes through three other singular points, although not as obvious as the previous ones: the midpoints of the segments that unite the three vertices with the orthocenter (D, F, H).

It is the circumference of the nine points, also known as Feuerbach’s circumference, for attributing its discovery to the German mathematician Karl Wilhelm Feuerbach (1800-1834), brother of the famous philosopher Ludwig Feuerbach, one of the parents of historical materialism.

The demonstration of the existence of Feuerbach’s circumference, as well as the determination of its center, is something complicated, but does not require more knowledge than those of elementary geometry, do you dare to try? And if you think it’s too difficult, you can start with a simpler one: demonstrate that Feuerbach’s circumference of a triangle is homothetic of its circumscribed circumference, and that the center of Homotecia is the orthocenter of the triangle. What is the reason for homotecia?

Recall that Homotecia is a geometric transformation with respect to a point (p) called Homotecia center, which makes it correspond to a similar figure (larger or smaller) such that at any point of the first (a, b, c) it corresponds to another of the second (a ‘, b’, c ‘) such that the reason for the distances of the homologous points to the center, called homotecia (k)

Pa/pa ‘= pb/pb’ = pc/pc ‘= k

Euler’s line

As Leonhard Euler demonstrated in 1765, the circumcentro, the orthocenter and the baricenter of a triangle are collinear, that is, they are on the same line, called Euler straight in honor of its discoverer. In addition, the center of Feuerbach’s circumference is also found on the same line (can you find where exactly?). And in some cases (in which?) The incenter is also found on Euler’s line.

And the thing doesn’t end there. Another point of Euler’s line that deserves a its own name is that of Longchamps, called in honor of the French mathematician Gaston de Longchamps (1842-1906), which is the symmetrical of the Ortocenter with respect to the circumcentro (can you see what makes it interesting?).

And if we complicate a little more the criteria for determining notable points of the Euler line, we will find Schiffler’s point: given an ABC triangle and its incentive I, its point of Schiffler is the one in which the respective straight lines of Euler of the triangles ABC, ABI, ACI and BCI (can you determine its exact location?

And there are more, such as Exeter’s point (discovered in a computer mathematics workshop) or the Far-out point, which will have to be left for another installment. Or several, because the special points of the apparently single polygon of three sides are so many that there is even an encyclopedia of notable points of the triangle.