Exeter point

In geometry , the Exeter point is one of the excellent points of a plane triangle . The Exeter Point has the designation X in Clark Kimberling's Encyclopedia of Triangle Centers (22). The point was discovered at a Phillips Exeter Academy computational math workshop in 1986. This makes it one of the triangular centers that have recently been found - in contrast to the classic points such as the center of gravity , the incircle center or the Steiner point .

definition

Exeter point of triangle ABC

The Exeter point is defined as follows:

Let an arbitrary triangle ABC be given. The points of intersection of the extended bisector with the circumference are denoted by A ' , B' and C ' . DEF is the triangle that is formed by the tangents to the circumference at points A , B and C ( D opposite A , E opposite B and F opposite C ). Then the straight lines DA ' , EB' and FC ' intersect at a point. This intersection is the Exeter point of triangle ABC .

Trilinear and barycentric coordinates

The trilinear coordinates of the Exeter point are

${\displaystyle (a(b^{4}+c^{4}-a^{4}),b(c^{4}+a^{4}-b^{4}),c(a^{4}+b^{4}-c^{4})),}$

the barycentric coordinates

${\displaystyle (a^{2}(b^{4}+c^{4}-a^{4}),b^{2}(c^{4}+a^{4}-b^{4}),c^{2}(a^{4}+b^{4}-c^{4})).}$

properties

• The Exeter point of triangle ABC lies on the Euler straight line of triangle ABC (the line that goes through the center of gravity , the vertical intersection and the circumcenter ).

Individual evidence

1. ^ Clark Kimberling: Encyclopedia of Triangle Centers: X (22) . Retrieved January 14, 2017.
2. ^ ^{A b} Clark Kimberling: Exeter Point . Retrieved January 14, 2017.
3. ^ Clark Kimberling: Triangle centers . Retrieved January 14, 2017.
4. ↑ Eric W. Weisstein: Exeter Point . From MathWorld - A Wolfram Web Resource. Retrieved January 14, 2017.