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How to prove that in an isosceles triangle circumcenter, centroid, orthocenter & incentre are collinear?

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    Any point definable from the triangle in terms of Euclidean geometry will be fixed by any Euclidean symmetry of the triangle.2011-02-13

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For each one, go back to the definition and prove it lies on the axis of symmetry. If you work with coordinates, putting the triangle at (-1,0), (1,0), (0,a) can make it much easier.

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As suggested in comments, consider the types of lines that determine each point (perpendicular bisectors of sides, medians, altitudes, and angle bisectors) and how these are affected by the triangle being isosceles. In particular, consider the symmetry that comes from the triangle being isosceles.

In addition, it's worth noting that the circumcenter, centroid, and orthocenter of a triangle are always collinear. As long as the triangle is not equilateral, they determine a line, which is called the Euler line of the triangle.