2.3  Exeter Point

Euler Line:

We have taken a look at four major centers of a triangle Orthocenter, Centroid, Circumcenter, and Incenter.  It was pointed out the three of these points the Orthocenter, Centroid, & Circumcenter or on the same line.  This line is called the Euler Line.  Different center of triangles can be found along this line.  The most recent point found on the line is the Exeter Point

Exeter Point of a Triangle:
The Exeter point is a triangle center. It was discovered in a "computers-in-mathematics" workshop at Phillips Exeter Academy in 1986. Unlike the classical triangle centers Orthocenter, Centroid, Circumcenter, and Incenter, the Exeter Point is a recent discover found by the use of a computer

Example

Build the Exeter Point at point E of ΔABC in Geogebra.

Solution: Step: 1.  Create ΔABC ( use black ) 3.  Create the Orthocenter of ΔABC      - Use the "Caption" to label the point         Orthocenter      - Hide the lines you made 4.  Create the Centroid of ΔABC      - Create a midpoint on each segment,           - Midpoint X on segment BC           - Midpoint Y on segment AC           - Midpoint Z on segment AB      - Create the following Rays,           - Ray AX. Ray BY, & Ray CZ      - The intersection of the rays will be the Centroid      - Use the "Caption" to label the point         Centroid      - Hide the lines you made 5.  Create the Circumcenter of ΔABC      - Create circle for the Circumcenter      - Use the "Caption" to label the point         Circumcenter      - Hide the lines you made 6.  Create Intersection Points with the Rays       and the Circle.      - Point A' at the intersection of         Ray AX and the Circle      - Point B' at the intersection of         Ray BY and the Circle      - Point C' at the intersection of         Ray CZ and the Circle 7.  Create a Tangent Line to the Circle at      - Point A, Point B, & Point C 8.   Create Intersection Points with the        Tangent Lines      - Point D at the intersection of         Tangent Lines B & C      - Point E at the intersection of         Tangent Lines A & C      - Point F at the intersection of         Tangent Lines A & B 9.  Create Intersecting Lines to find the       Exeter Point      - Lines DA', EB', & FC'      - Put the Exeter Point at the intersection        of these lines. 10.  Create the Euler Line using any two of        the following points      -Orthocenter      -Centroid      -Circumcenter You will now see that Exeter Point lies on the same line as the other 3-poijnts Press or to run the Geogebra build simulation.