Increasing knowledge: the five thousand hearts of the triangle

After learning plane geometry in middle and high school, you must know the center of gravity, circumcenter, orthocenter, incenter and paracenter of a triangle, which are commonly known as the five centers of a triangle.

Does a triangle only have five centers? In fact, these five hearts are just the tip of the iceberg.

The center of a triangle (Triangle center) refers to a series of points on the plane. To a certain extent, they are similar to the centers of squares or circles, which are the centers of triangles. How many points are there in this series? Currently, there are 5,389. Although it is a lot, not all of them are sufficient.

If we want to discuss the heart of a triangle, we must start with a night in ancient Greece 2,000 years ago. When the mathematicians there got tired of looking up at the stars, they lowered their heads and drew a triangle. They accidentally discovered the hearts of some classic triangles, such as the five hearts mentioned at the beginning. Of course, they did not define what the center of a triangle is, nor did they specify what kind of point could be called the center of a triangle. After them, some centroids of triangles were discovered one after another, such as Fermat points, first equal angle points, Gergonne points, Weibit points, Gergonne points, Feuerbach points, etc. By the 1980s, there was a resurgence of interest in trigonometric geometry, and it was noted that these points shared some of the same properties. On this basis, the center of the triangle is clearly defined. In October 2012, a mathematician named Clark Kimberling released an online encyclopedia of triangles compiled by him, which contains 5389 points. It is quite troublesome to name these points one by one, so they are numbered X (1), X (2)...