Nine-point center

In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle.

It is so called because it is the center of the nine-point circle (also known as Feuerbach's circle after Karl Wilhelm Feuerbach), a circle that passes through nine significant points of the triangle: the midpoints of the three edges, the feet of the three altitudes, and the points halfway between the orthocenter and each of the three vertices. The nine-point center is listed as point X(5) in Clark Kimberling's Encyclopedia of Triangle Centers.

Properties

The nine-point center N lies on the Euler line of its triangle, at the midpoint between that triangle's orthocenter H and circumcenter O. The centroid G also lies on the same line, 2/3 of the way from the orthocenter to the circumcenter, so

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{\displaystyle |NO|=|NH|=3|NG|.}

Thus, if any two of these four triangle centers are known, the positions of the other two may be determined from them.

Andrew Guinand proved in 1984, as part of what is now known as Euler's triangle determination problem, that if the positions of these centers are given for an unknown triangle, then the incenter of the triangle lies within the orthocentroidal circle (the circle having the segment from the centroid to the orthocenter as its diameter). The only point inside this circle that cannot be the incenter is the nine-point center, and every other interior point of the circle is the incenter of a unique triangle.

The distance from the nine-point center to the incenter I satisfies

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{\displaystyle {\begin{aligned}&|IN|<{\tfrac {1}{2}}|IO|,\\&|IN|={\tfrac {1}{2}}(R-2r)<{\frac {R}{2}},\\&2R\cdot |IN|=|OI|^{2},\end{aligned}}}

where R, r are the circumradius and inradius respectively.

The nine-point center is the circumcenter of the medial triangle of the given triangle, the circumcenter of the orthic triangle of the given triangle, and the circumcenter of the Euler triangle. More generally it is the circumcenter of any triangle defined from three of the nine points defining the nine-point circle.