geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the GEOS circle is derived from the intersection of four lines that are associated with a generalized

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

: the

Euler line In geometry, the Euler line, named after Leonhard Euler ( ), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, incl ...

, the

Soddy line The Soddy line of a triangle is the line that goes through the centers of the two Soddy circles of that triangle. The Soddy line intersects the Euler line in the de Longchamps point and the Gergonne line in the ''Fletcher point''. It is also perp ...

, the orthic axis and the

Gergonne line Joseph Diez Gergonne (19 June 1771 at Nancy, France – 4 May 1859 at Montpellier, France) was a French mathematician and logician. Life In 1791, Gergonne enlisted in the French army as a captain. That army was undergoing rapid expansion becaus ...

. Note that the Euler line is

orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...

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 In geometry, an orthocentric system is a set (mathematics), set of four point (geometry), points on a plane (mathematics), plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through ...

. The GEOS circle is that circle centered at a point equidistant from ''X''650 (the intersection of the orthic axis with the Gergonne line) and ''X''20 (the intersection of the Euler line with the Soddy line and is known as the

de Longchamps point In geometry, the de Longchamps point of a triangle is a triangle center named after French mathematician Gaston Albert Gohierre de Longchamps. It is the reflection of the orthocenter of the triangle about the circumcenter.. Definition Let th ...

) and passes through these points as well as the two points of orthogonal intersection. The orthogonal intersection points are ''X''468 (the intersection of the orthic axis with the Euler line) and ''X''1323 (the

Fletcher point Fletcher may refer to: People and fictional characters * Fletcher (surname), including lists of people and fictional characters * Fletcher (given name), lists of people and fictional characters * Fletcher (occupation), a person who fletches arr ...

, the intersection of the Gergonne line with the Soddy line). 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). The ''X''(''i'') point notation is the Clark Kimberling

ETC * Etc. or et cetera, a Latin expression meaning "and the other things" or "and the rest". ETC or etc may also refer to: Companies and organizations * ETC (Chilean TV channel), a Chilean cable television channel * ETC (Philippine TV channel), a ...

classification of

centers.

References

* Circles defined for a triangle {{elementary-geometry-stub