What type of triangle has its circumcenter, its incenter, its centroid, and its orthocenter all at the same point?

The only triangle with the usual triangle centers coincident is an equilateral triangle.

The centroid is the meet of the medians, which are the segments from a vertex to the midpoint of its opposite side. The centroid is the balance point of the triangle.  It always divides the medians into a 2:1 ratio, the long piece connected to the vertex.

The circumcenter is the center of the circumcircle, which is the circle through the vertices.  That's only going to be the centroid when that 2 part of the 2:1 ratio is the same for each median.  That's only going to happen when the sides are the same length so the centroid makes congruent triangles.

The incenter is the center of the inscribed circle, which is the meet of the angle bisectors of each vertex.  Again it wont be the centroid unless the sides are equal.

The orthocenter is the meet of the altitudes and it's the same story.

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step-by-step explanation: