Editing Orthocentric system (section)

==The common orthic triangle, its incenter, and its excenters==

If the six connectors that join any pair of orthocentric points are extended to six lines that intersect each other, they generate seven intersection points. Four of these points are the original orthocentric points and the additional three points are the [[orthogonal]] intersections at the feet of the [[altitude]]s. The joining of these three orthogonal points into a triangle generates an [[orthic triangle]] that is common to all the four possible triangles formed from the four orthocentric points taken three at a time.

The [[incenter]] of this common orthic triangle must be one of the original four orthocentric points. Furthermore, the three remaining points become the [[excenter]]s of this common orthic triangle.  The orthocentric point that becomes the incenter of the orthic triangle is that orthocentric point closest to the common nine-point center. This relationship between the orthic triangle and the original four orthocentric points leads directly to the fact that the incenter and excenters of a reference triangle form an orthocentric system.{{sfn|Johnson|1929|p=[https://babel.hathitrust.org/cgi/pt?id=wu.89043163211&view=1up&seq=200 182]}}

It is normal to distinguish one of the orthocentric points from the others, specifically the one that is the incenter of the orthic triangle; this one is denoted {{mvar|H}} as the orthocenter of the outer three orthocentric points that are chosen as a reference triangle {{math|△''ABC''}}. In this normalized configuration, the point {{mvar|H}} will always lie within the triangle {{math|△''ABC''}}, and all the angles of triangle {{math|△''ABC''}} will be acute. The four possible triangles referred above are then triangles {{math|△''ABC'', △''ABH'', △''ACH'', △''BCH''}}. The six connectors referred above are {{mvar|{{overline|AB}}, {{overline|AC}}, {{overline|BC}}, {{overline|AH}}, {{overline|BH}}, {{overline|CH}}}}. The seven intersections referred above are {{mvar|A, B, C, H}} (the original orthocentric points), and {{mvar|H{{sub|A}}, H{{sub|B}}, H{{sub|C}}}} (the feet of the altitudes of triangle {{math|△''ABC''}} and the vertices of the orthic triangle).