Exeter point

[[File:Exeter point.svg|thumb|300px|

{{legend-line|solid black|Reference triangle {{math|△ABC}}}}

{{legend-line|dashed black 2px|Medians of {{math|△ABC}}; concur at centroid}}

{{legend-line|solid cyan|Circumcircle of {{math|△ABC}}}}

{{legend-line|solid #2fc437|Triangle {{math|△A'B'C' }} formed by intersection of medians with circumcircle}}

{{legend-line|solid magenta|Tangential triangle {{math|△DEF}} of {{math|△ABC}}}}

{{legend-line|solid red|Lines joining vertices of {{math|△DEF}} and {{math|△A'B'C' }}; concur at Exeter point}}

]]

The Exeter point is defined as follows.{{cite web|last=Weisstein|first=Eric W.|title=Exeter Point|url=http://mathworld.wolfram.com/ExeterPoint.html|publisher=From MathWorld--A Wolfram Web Resource|accessdate=24 May 2012}}

:Let {{math|△ABC}} be any given triangle. Let the medians through the vertices {{mvar|A, B, C}} meet the circumcircle of {{math|△ABC}} at {{mvar|A', B', C'}} respectively. Let {{math|△DEF}} be the triangle formed by the tangents at {{mvar|A, B, C}} to the circumcircle of {{math|△ABC}}. (Let {{mvar|D}} be the vertex opposite to the side formed by the tangent at the vertex {{mvar|A}}, {{mvar|E}} be the vertex opposite to the side formed by the tangent at the vertex {{mvar|B}}, and {{mvar|F}} be the vertex opposite to the side formed by the tangent at the vertex {{mvar|C}}.) The lines through {{mvar|DA', EB', FC'}} are concurrent. The point of concurrence is the Exeter point of {{math|△ABC}}.

The trilinear coordinates of the Exeter point are

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

• 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.

{{reflist}}{{Phillips Exeter Academy}}

Category:Triangle centers