GEOS circle
Encyclopedia
In geometry
, the GEOS circle is derived from the intersection of four lines that are associated with a generalized triangle
: the Euler line, the Soddy line, the orthic axis and the Gergonne line. Note that the Euler line is orthogonal to the orthic axis and that the Soddy line is orthogonal to the Gergonne line.
These four lines provide six points of intersection of which two points occur at line intersections that are orthogonal. Consequently the other four points form an orthocentric system
.
The GEOS circle is that circle centered at a point equidistant from X650 (the intersection of the orthic axis with the Gergonne line) and X20 (the intersection of the Euler line with the Soddy line and is known as the de Longchamps point
) and passes through these points as well as the two points of orthogonal intersection.
The orthogonal intersection points are X468 (the intersection of the orthic axis with the Euler line) and X1323 (the intersection of the Gergonne line with the Soddy line and is known as the Fletcher point).
The orthocentric system comprises X650, X20, X1375 (the intersection of the Euler line with the Gergonne line and is known as the Evans point
) and X3012 (the intersection of the Soddy line and the orthic axis).
The X(i) point notation is the Clark Kimberling ETC
classification of triangle
centers.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the GEOS circle is derived from the intersection of four lines that are associated with a generalized triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
: the Euler line, the Soddy line, the orthic axis and the Gergonne line. Note that the Euler line is orthogonal to the orthic axis and that the Soddy line is orthogonal to the Gergonne line.
These four lines provide six points of intersection of which two points occur at line intersections that are orthogonal. Consequently the other four points form an orthocentric system
Orthocentric system
In geometry, an orthocentric system is a set of four points in the plane one of which is the orthocenter of the triangle formed by the other three....
.
The GEOS circle is that circle centered at a point equidistant from X650 (the intersection of the orthic axis with the Gergonne line) and X20 (the intersection of the Euler line with the Soddy line and is known as the de Longchamps point
De Longchamps point
In geometry, the de Longchamps point of a triangle is the reflection of its orthocenter about its circumcenter. It is listed as X in the Encyclopedia of Triangle Centers. Its trilinear coordinates are\displaystyle\cos A - \cos B \cos C : \cos B - \cos C \cos A : \cos C - \cos A \cos BThe point is...
) and passes through these points as well as the two points of orthogonal intersection.
The orthogonal intersection points are X468 (the intersection of the orthic axis with the Euler line) and X1323 (the intersection of the Gergonne line with the Soddy line and is known as the Fletcher point).
The orthocentric system comprises X650, X20, X1375 (the intersection of the Euler line with the Gergonne line and is known as the Evans point
Evans Point
Evans Point is an ice-covered point fronting on Peacock Sound, lying 15 nautical miles west-northwest of Von der Wall Point on the south side of Thurston Island. First plotted from air photos taken by U.S. Navy Operation Highjump in December 1946...
) and X3012 (the intersection of the Soddy line and the orthic axis).
The X(i) point notation is the Clark Kimberling ETC
Encyclopedia of Triangle Centers
The Encyclopedia of Triangle Centers is an on-line list of more than 3,000 points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville....
classification of triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
centers.