In the last two weeks we have talked about: 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, happiness 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 OrtoTenter (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 (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 it does not require more knowledge than those of, 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 transformation with respect to a point (p) called Homotecia center, which corresponds to a similar figure (larger or smaller) such that at full 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 reason (k), be constant:
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 on Euler’s line that deserves 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 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 the point of Schiffler: given a and its incentive I, its point of Schiffler is one in which the respective lines of Euler of the triangles ABC, ABI, ACI and BCI (can you determine its exact location, or when less approximate?).
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.
