definition
A point of concurrency is the point where three or more lines intersect.

The term "concurrent", or "concurrency", is commonly seen in geometry in relation to triangles.

Ancient Greek mathematicians knew of the existence of specific lines, which when drawn in relation to a triangle, could intersect in one common point inside the triangle. When the first such line was discovered, mathematicians thought that they had discovered the "center" of the triangle.
Upon further investigation, however, it was discovered that there were other specific lines that also intersected in one common point inside the triangle, but these common points did not coincide with the previously found "center" point. All of the discovered common points were deemed to be "triangle centers" in the sense that each of them could be described as the center of the triangle under certain conditions.

So, how many "centers" does a triangle possess? The Greeks knew of three such "centers", the centroid, the orthocenter, and the circumcenter, referred to as the classical triangle centers. Over time, mathematicians have extended the list of "triangle centers" to contain over 100 entries.

It was not until the eighteenth century that mathematician Leonhard Euler (1707-1783) discovered a relationship between the three classical triangle centers. Euler discovered that in any triangle, the centroid, orthocenter and circumcenter are collinear (they lie on the same straight line).

In addition, the centroid is always located between the orthocenter and the circumcenter.

The distance from the centroid to the orthocenter is always twice the distance from the centroid to the circumcenter.

The line upon which these three classical triangle centers are collinear is called the Euler line.

Note: The name "Euler" is pronounced as if it were spelled "Oiler" in English.
EulerLine

We will start our investigation with the three classical triangle centers: the centroid, the orthocenter, and the circumcenter. We will also investigate the incenter.

The ancient Greek mathematicians concentrated on these "triangle centers" being located "within" the interior of the triangle. We will also look at which of these "centers" may be located "outside", or "on", the triangle.



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