ninepointsirkelen

The ninepointsirkelen, or the nine-point circle, is a circle in Euclidean geometry associated with a triangle. It is also known as Euler's circle.

It passes through nine significant points: the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments from each vertex to the orthocenter.

The center of the nine-point circle is the nine-point center, denoted N, which is the midpoint of the segment OH where O is the circumcenter and H is the orthocenter. The circle has radius R/2, where R is the circumradius of the triangle.

The nine-point circle is the circumcircle of the medial triangle (the triangle formed by the midpoints of the sides). It is related to the Euler line: the line through the circumcenter O and the orthocenter H also contains the nine-point center N, which sits at the midpoint of OH.

Feuerbach's theorem states that the nine-point circle is tangent to the incircle and to the excircles of the triangle; the points of tangency are known as Feuerbach points.

The concept was rediscovered and popularized by Euler in the 18th century and remains a standard construct in triangle geometry, appearing in many proofs and geometric constructions.