Editing Orthocentric system (section)

==The common nine-point circle==
{{further|Nine-point circle}}
[[File:Orthocentric system nine point circle.svg|thumb|upright=1.0|'''Common nine-point circle''', where {{math|''N, O''{{sub|4}}, ''A''{{sub|4}}}} are the nine-point center, circumcenter, and orthocenter respectively of the triangle formed from the other three orthocentric points {{math|''A''{{sub|1}}, ''A''{{sub|2}}, ''A''{{sub|3}}}}.]]

The center of this common nine-point circle lies at the [[centroid]] of the four orthocentric points. The radius of the common nine-point circle is the distance from the nine-point center to the midpoint of any of the six connectors that join any pair of orthocentric points through which the common nine-point circle passes. The nine-point circle also passes through the three orthogonal intersections at the feet of the altitudes of the four possible triangles.

This common nine-point center lies at the midpoint of the connector that joins any orthocentric point to the circumcenter of the triangle formed from the other three orthocentric points.

The common nine-point circle is tangent to all 16 incircles and excircles of the four triangles whose vertices form the orthocentric system.<ref>Weisstein, Eric W. "Orthocentric System." From MathWorld--A Wolfram Web Resource. [http://mathworld.wolfram.com/OrthocentricSystem.html]</ref>