In geometry, the Exeter point is a special point associated with a

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

triangle. The Exeter point is a triangle center and is designated as the center X(22) in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a computers-in-mathematics workshop at

Phillips Exeter Academy (not for oneself) la, Finis Origine Pendet (The End Depends Upon the Beginning) gr, Χάριτι Θεοῦ (By the Grace of God) , location = 20 Main Street , city = Exeter, New Hampshire , zipcode ...

in 1986. This is one of the recent triangle centers, unlike the classical triangle centers like

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any o ...

, incenter, and Steiner point.

Definition

Exeter point is defined as follows. :Let ''ABC'' be any given triangle. Let the

medians The Medes (Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, the ...

through the vertices ''A'', ''B'', ''C'' meet the

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

of triangle ''ABC'' at ''A' '', ''B' '' and ''C' '' respectively. Let ''DEF'' be the triangle formed by the tangents at ''A'', ''B'', and ''C'' to the circumcircle of triangle ''ABC''. (Let ''D'' be the vertex opposite to the side formed by the tangent at the vertex ''A'', ''E'' be the vertex opposite to the side formed by the tangent at the vertex ''B'', and ''F'' be the vertex opposite to the side formed by the tangent at the vertex ''C''.) The lines through ''DA' '', ''EB' '' and ''FC' '' are

concurrent Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea'' * Concurring opinion (also called a "concurrence"), a ...

. The point of concurrence is the ''Exeter point'' of triangle ''ABC''.

Trilinear coordinates

The

trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...

of the Exeter point are : ( ''a'' ( ''b''⁴ + ''c''⁴ − ''a''⁴ ), ''b'' ( ''c''⁴ + ''a''⁴ − ''b''⁴ ), ''c'' ( ''a''⁴ + ''b''⁴ − ''c''⁴ ) ).

Properties

*The Exeter point of triangle ''ABC'' lies on 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, includ ...

(the line passing through the

, the orthocenter , the de Longchamps point, the Euler centre and the

circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

) of triangle ''ABC''. So there are 6 points collinear over one only line.

References

{{Phillips Exeter Academy Triangle centers