Centers of Common Triangles - Expii

For example, the circumcenter, incenter, centroid, and orthocenter are all the same for an equilateral triangle. For an isosceles triangle, they all lie on the line of symmetry of the isosceles triangle. On the other hand, for a right triangle, the circumcenter is the midpoint of the hypotenuse, and the orthocenter is the right-angled vertex (i.e. the vertex opposite the hyponuse).