Notable points

The notable points of the triangle , which we discussed last week, have, excuse the redundancy, aroused considerable interest among my kind readers, which encourages me to continue exploring the topic. It is easy to show that the three perpendicular bisectors of the sides of a triangle intersect at a point.

Given a triangle ABC, all points on the perpendicular bisector of side AB are equidistant from A and B, and all points on the perpendicular bisector of side AC are equidistant from A and C; therefore, the point of intersection of both perpendicular bisectors is also equidistant from B and C, so the perpendicular bisector of BC must pass through it. And since this point is equidistant from the three vertices of the triangle, it is the center of the circle that passes through them, that is, the circumcenter.

The demonstration that the angle bisectors meet at a point is analogous: all points of the angle bisector of angle A are equidistant from the sides AB and AC; all points of the angle bisector of angle B are equidistant from the sides AB and BC; therefore, the point of intersection of both angle bisectors will also be equidistant from the sides AC and BC, and therefore the angle bisector of angle C will pass through it. And since this point is equidistant from the three sides of the triangle, it is the center of the circle tangent to all three, that is, the incenter.

The demonstration that medians meet at a point is not so simple. I found Bretos Bursó 's recursive proposal interesting: "Let T0 be the initial triangle. The midpoints of each side are the vertices of a triangle T1 similar to T0 by Thales's theorem, and by the same theorem, each median of T0 contains a median of T1. One can iterate indefinitely by considering a sequence of similar triangles Tn decreasing in content—now I call a triangle, in each case, the compact one that includes the interior. In all triangles, the medians are segments of the initial ones; and, by Cantor's theorem, the intersection of all of them is a single-point set whose element must lie on the three medians. That point is the centroid."

And, as Manuel Amorós says: "A physical way to determine the centroid of a triangle could be to hang a weighted thread. At one point on this thread, we hang the triangle's sheet by one of its vertices. The thread that crosses the sheet will mark a median. We perform the same procedure hanging from another vertex, and we obtain two medians, whose intersection will be the centroid." And the procedure also applies to a pentagon or any other figure.

Regarding the orthocenter, to demonstrate that the altitudes of a triangle meet at a point, Euclid drew a straight line parallel to the opposite side through each vertex. The three lines, when intersected, form a triangle whose perpendicular bisectors are the altitudes of the original triangle, and as we have already demonstrated that the perpendicular bisectors meet at a point…

The Feuerbach circle

A newly added commentator, María Beatriz Collado, says: “I love reading a newspaper article about the essential points of a triangle . Will you tell us about the 9-point circle after this, which is the most wonderful thing I've ever encountered in geometry?”

Well, yes, it's a wonderful topic that I'll be dedicating a future post to. Since I'm running out of time for this one, I can only state it (and announce it):

The 9-point circle is the circle that, in any triangle, passes through the midpoints of the three sides, the bases of the three altitudes, and the midpoints of the segments that join the three vertices to the orthocenter of the triangle. It is also known as the Feuerbach circle. But, as I said, that will be a separate article.