Proof of the existence of the Euler line


Let C, G be the Circumcenter and Centroid. Produce CG to H so that CG = one half GH.

It remains to show that H is the Orthocenter.
Join AH and produce it to meet BD at P. Since

and therefore AH is parallel to QC, but CQ is perpendicular to BD and so AP is an altitude of the triangle. In a similar way DH extended is perpendicular to AB and H is the orthocenter.



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