The circumcubic known as the orthocubic-K006 is interesting in that it passes through three orthocentric systems as well as the three vertices of the orthic triangle ( but not the orthocenter of the orthic triangle ).
32.
So if a rectangular hyperbola is drawn through four orthocentric points it will have one fixed center on the common nine-point circle but it will have four perspectors one on each of the orthic axes of the four possible triangles.
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The three orthocentric systems are the incenter and excenters, the reference triangle and its orthocenter and finally the orthocenter of the reference triangle together with the three other intersection points that this cubic has with the circumcircle of the reference triangle.
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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.
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The orthic axis associated with a normalized orthocentric system A, B, C and H, where ABC is the reference triangle, is a line that passes through three intersection points formed when each side of the orthic triangle meets each side of the reference triangle.
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The Euler lines of the four triangles formed by an orthocentric system ( a set of four points such that each is the orthocenter of the triangle with vertices at the other three points ) are concurrent at the nine-point center common to all of the triangles.
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"' Common nine-point circle "', where O, O 4 and A 4 are the nine-point center, circumcenter, and orthocenter respectively of the triangle formed from the other three orthocentric points A 1, A 2 and A 3.
38.
The orthocentric system comprises " X " 650, " X " 20, " X " 1375 ( the intersection of the Euler line with the Gergonne line and is known as the Evans point ) and " X " 3012 ( the intersection of the Soddy line and the orthic axis ).
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It can be also extended from plane Euclidean geometry to higher dimension Euclidean spaces ( e . g ., to tetrahedra and more generally to simplices ), as has been done for orthocentric tetrahedra ( i . e ., tetrahedra in which altitudes are concurrent ) and more generally for orthocentric simplices ( i . e ., simplices in which altitudes are concurrent ).
40.
It can be also extended from plane Euclidean geometry to higher dimension Euclidean spaces ( e . g ., to tetrahedra and more generally to simplices ), as has been done for orthocentric tetrahedra ( i . e ., tetrahedra in which altitudes are concurrent ) and more generally for orthocentric simplices ( i . e ., simplices in which altitudes are concurrent ).
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