Exeter point

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

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The Exeter point is defined as follows.

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

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

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

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 on a single line.