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Exeter point

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Triangle center

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) in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a computers-in-mathematics workshop at Phillips Exeter Academy in 1986. This is one of the recent triangle centers, unlike the classical triangle centers like centroid, incenter, and Steiner point.

Definition

  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.

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.

Trilinear coordinates

The trilinear coordinates 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 ) {\displaystyle a(b^{4}+c^{4}-a^{4}):b(c^{4}+a^{4}-b^{4}):c(a^{4}+b^{4}-c^{4})}

Properties

References

  1. Kimberling, Clark. "Encyclopedia of Triangle Centers: X(22)". Retrieved 24 May 2012.
  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.
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