The centroid G,the orthocenter H, and the circumcenter C lies on a unique line and the line through G,H,O is called the Euler line of the triangle.

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Consider several shapes of triangles.

If triangle ABC isn't equilateral, the point G, H, C are distinct, lie on a euler line, and satisfy GH/GO=-2.

The relation GH/GO=-2 shows that G lies between H and C, twice as far from C. Thus G lies one-third of the way from C to H.

If triangle ABC is equilateral, the points G, H, C, and I are equal.

All three sides of triangle ABC have the same length; so the three altitudes, the three medians an dthe three perpendicular bisectors are all equal. th ethree medians interect at the unique point G, and the three perpendicular bisectors intersect at the unique point C, and so the three altitudes intersect at a unique point H, where G=H=C.

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