ninepoint

Ninepoint, often written as ninepoint, most commonly refers to the nine-point circle in triangle geometry, also known as Euler's circle. This circle passes through nine significant points associated with a given triangle: the midpoints of its three sides, the feet of its three altitudes, and the midpoints of the segments from each vertex to the orthocenter.

Geometric properties include that these nine points are concyclic on the nine-point circle. The center of this circle, called the nine-point center, lies on the Euler line and is the midpoint of the segment joining the triangle’s orthocenter and its circumcenter. The radius of the nine-point circle is half the triangle’s circumradius, r9 = R/2. The circle is also the incircle of the medial triangle, which is formed by the midpoints of the sides.

Relation to broader geometry: The nine-point circle exists for all triangle types—acute, obtuse, and right—though certain points may coincide in special cases (for example, in a right triangle the orthocenter lies at a vertex, causing some redundancy among the nine points). The concept is closely linked to other triangle centers and lines, notably the Euler line that connects the circumcenter, centroid, and orthocenter.

Other usages: Outside geometry, the term ninepoint may be adopted as a brand name or project designation in various fields. In such cases, it is a proper noun distinct from the geometric concept described above.

See also: nine-point circle, Euler line, triangle centers.