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The points G and I remain inside the triangle at all times. The centroid (G) remains inside because for all triangles, the medians are always contained inside the triangle. The incenter (I) remains inside because the only way angle bisectors could converge outside the traingle would be if one of the bisectors was larger than the angle it bisects. By the definition of bisector, this is not possible.

The points H and C are located outside the triangle when it is obtuse and inside when it is acute. They are located on the triangle when it is a right traingle. We have shown that the circumcenter is on the midpoint of the hypotenuse of a right triangle and the orthocenter is on the vertex across from the hypotenuse of a right triangle. Therefore, these are the positions through which H and C move outside the triangle, and they do so when the triangle is a right triangle.