Blame It On Euler-question

Name: Due Blame it on Euler The name Leonhard Euler (pronounced "Oiler") shows up quite often in the study of geometry. There is even a line named after him. The Euler line is the line that contains the orthocenter, the centroid, and the circumcenter of a triangle.

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Notes: The altitude of a triangle is a segment from a vertex perpendicular to the opposite side. The orthocenter is the point of intersection of the three altitudes of a triangle. The median of a triangle is a segment from a vertex to the midpoint of the opposite side. The centroid is the point of intersection of the three medians of a triangle. The perpendicular bisector of a segment is the line perpendicular to the segment at the midpoint of the segment. The circumcenter is the point of intersection of the perpendicular bisectors.

Find the equation of Euler's line for the triangle with vertices (! 10, ! 4), (! 6,8)and (10, ! 8) and show that the orthocenter, circumcenter and centroid are all on this line.

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Use a separate piece of paper and attach the problem as the cover sheet. Show all of your mathematics. Include any charts and graphs you used to solve the problem. Do not turn in your scratch paper. Organize your work sequentially so that it can be read start to finish rather than scattered over a page.

Need a greater challenge? Write a program on the graphing calculator or use a spreadsheet to determine Euler's line given any three points in the coordinate plane. (I needed to do this to search for a good combination of numbers to use on this challenge.)