Nine-point circle

The nine-point circle of a triangle passes through the midpoints of its sides, the feet of its altitudes, and the midpoints between each vertex and the orthocenter.

Orthocenter: The point where a triangle’s three altitudes intersect. Three of the circle’s points are the feet of the altitudes through the vertices.
Nine-point center: The center of a triangle’s nine-point circle, halfway between its circumcenter and orthocenter. This point organizes the circle’s position relative to the triangle’s principal centers.
Karl Wilhelm Feuerbach: A German mathematician known for proving the tangent relation between a triangle’s nine-point circle and its incircle. His 1822 result established the famous contact property associated with the circle.
Orthic triangle: The triangle formed by the feet of a triangle’s three altitudes. Its vertices are three of the nine points, making the nine-point circle its circumcircle.
Circumcenter: The center of the circle passing through a triangle’s three vertices. The nine-point center lies halfway between the circumcenter and orthocenter.
Nine-point radius: The radius of a triangle’s nine-point circle, equal to half its circumradius. Its fixed relation to the circumradius gives a compact metric description of the circle.
Olry Terquem: A French mathematician who studied triangle geometry and gave the circle its name. Terquem’s terminology helped establish the modern name for the circle.
Incircle: The circle inside a triangle tangent to all three sides. Feuerbach’s theorem states that it is tangent internally to the nine-point circle.
Midpoint: The point exactly halfway between two points on a line segment. Three side midpoints and three vertex–orthocenter midpoints lie on the circle.
Euler line: The line containing a triangle’s circumcenter, centroid, and orthocenter. The nine-point center lies on this line, midway between circumcenter and orthocenter.

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