Editing Euler line (section)

===Distances between centers===

On the Euler line the centroid ''G'' is between the circumcenter ''O'' and the orthocenter ''H'' and is twice as far from the orthocenter as it is from the circumcenter:<ref name="ac">Altshiller-Court, Nathan, ''College Geometry'', Dover Publications, 2007 (orig. Barnes & Noble 1952).</ref>{{rp|p.102}}

:<math>GH=2GO;</math>
:<math>OH=3GO.</math>

The segment ''GH'' is a diameter of the [[orthocentroidal circle]].

The center ''N'' of the nine-point circle lies along the Euler line midway between the orthocenter and the circumcenter:<ref name="k"/>

:<math>ON = NH, \quad OG =2\cdot GN, \quad NH=3GN.</math>

Thus the Euler line could be repositioned on a number line with the circumcenter ''O'' at the location 0, the centroid ''G'' at 2''t'', the nine-point center at 3''t'', and the orthocenter ''H'' at 6''t'' for some scale factor ''t''.

Furthermore, the squared distance between the centroid and the circumcenter along the Euler line is less than the squared [[circumradius]] ''R''<sup>2</sup> by an amount equal to one-ninth the sum of the squares of the side lengths ''a'', ''b'', and ''c'':<ref name="ac"/>{{rp|p.71}}

:<math>GO^2=R^2-\tfrac{1}{9}(a^2+b^2+c^2).</math>

In addition,<ref name="ac"/>{{rp|p.102}}

:<math>OH^2=9R^2-(a^2+b^2+c^2);</math>

:<math>GH^2=4R^2-\tfrac{4}{9}(a^2+b^2+c^2).</math>