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

The Exeter point is defined as follows.

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 .

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

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.

Exeter point

Definition

Trilinear coordinates

Properties

References