Enanäliitteen

Enanäliitteen, also known as the "nine-point circle" or "nine-point conic," is a notable circle in triangle geometry. It is named for the nine significant points it passes through, which are related to the triangle's vertices, altitudes, and midpoints. The concept was first introduced by the French mathematician Jean-Victor Poncelet in 1822, but it was later popularized by the German mathematician Karl Wilhelm Feuerbach, who proved many of its properties.

The enanäliitteen is defined for any given triangle. It passes through 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 intersection point of the altitudes). These nine points are concyclic, meaning they all lie on the same circle. The center of the enanäliitteen is the midpoint of the line segment connecting the orthocenter and the circumcenter (the center of the circumscribed circle of the triangle). The radius of the enanäliitteen is half the radius of the circumscribed circle.

The enanäliitteen has several important properties and is related to various other geometric constructs. For instance, it is tangent to the incircle and the three excircles of the triangle, a fact known as Feuerbach's theorem. This circle plays a significant role in triangle geometry and is a subject of ongoing research and exploration in mathematical circles.