Exeter point

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In geometry, the Exeter point is a special point associated with a plane triangle. It is a triangle center and is designated as 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, unlike the classical triangle centers like centroid, incenter, and Steiner point. [3]

Contents

Definition

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Reference triangle ^ABC
Medians of ^ABC; concur at centroid
Circumcircle of ^ABC
Triangle ^A'B'C' formed by intersection of medians with circumcircle
Tangential triangle ^DEF of ^ABC
Lines joining vertices of ^DEF and ^A'B'C' ; concur at Exeter point Exeter point.svg
  Reference triangle ABC
   Medians of ABC; concur at centroid
   Circumcircle of ABC
  Triangle A'B'C' formed by intersection of medians with circumcircle
   Tangential triangle DEF of ABC
  Lines joining vertices of DEF and A'B'C' ; concur at Exeter point

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 ABC at A', B', C' respectively. Let DEF be the triangle formed by the tangents at A, B, C to the circumcircle of 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', FC' are concurrent. The point of concurrence is the Exeter point of ABC.

Put succinctly, the Exeter point is the perspector of the circummedial triangle and the tangential triangle.

Trilinear coordinates

The trilinear coordinates of the Exeter point are

Properties

References

  1. Kimberling, Clark. "Encyclopedia of Triangle Centers: X(22)" . Retrieved 24 May 2012.
  2. 1 2 Kimberling, Clark. "Exeter Point" . Retrieved 24 May 2012.
  3. Kimberling, Clark. "Triangle centers" . Retrieved 24 May 2012.
  4. Weisstein, Eric W. "Exeter Point". From MathWorld--A Wolfram Web Resource. Retrieved 24 May 2012.