Exeter point explained

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]

Definition

The Exeter point is defined as follows.^{[2] [4]}

Let be any given triangle. Let the medians through the vertices meet the circumcircle of at respectively. Let be the triangle formed by the tangents at to the circumcircle of . (Let be the vertex opposite to the side formed by the tangent at the vertex, be the vertex opposite to the side formed by the tangent at the vertex, and be the vertex opposite to the side formed by the tangent at the vertex .) The lines through are concurrent. The point of concurrence is the Exeter point of .

Trilinear coordinates

The trilinear coordinates of the Exeter point are

$$a(b^4\; +\; c^4\; -\; a^4)\; :\; b(c^4\; +\; a^4\; -\; b^4)\; :\; c(a^4\; +\; b^4\; -\; c^4)$$

Properties

• The Exeter point of triangle ABC lies on the Euler line (the line passing through the centroid, the orthocenter, the de Longchamps point, the Euler centre and the circumcenter) of triangle ABC. So there are 6 points collinear over one only line.

Notes and References

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