geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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 (gen ...

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

. The Exeter point is a

triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example ...

and is designated as the center X(22) in

Clark Kimberling Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer seq ...

Encyclopedia of Triangle Centers The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the l ...

. 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 ob ...

incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...

, 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, th ...

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, inclu ...

(the line passing through the

, the

orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...

, 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 the ...

, the

Euler centre Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

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