Exeter point
From Infogalactic: the planetary knowledge core
In geometry, the Exeter point is a special point associated with a plane triangle. The Exeter point is a triangle center and is designated as the center X(22)[1] in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a computers-in-mathematics workshop at Phillips Exeter Academy in 1986.[2] This is one of the recent triangle centers, having been discovered only in 1986, unlike the classical triangle centers like centroid, incenter, and Steiner point.[3]
Definition
File:Exeter point.svg
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The Exeter point is defined as follows.[2][4]
- Let ABC be any given triangle. Let the medians through the vertices A, B, C meet the circumcircle 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. The point of concurrence is the Exeter point of triangle ABC.
Trilinear coordinates
The trilinear coordinates of the Exeter point are
- ( a ( b4 + c4 − a4 ), b ( c4 + a4 − b4 ), c ( a4 + b4 − c4 ) ).
Properties
- The Exeter point of triangle ABC lies on the Euler line (the line passing through the centroid, the orthocenter and the circumcenter) of triangle ABC.
