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Circumcenter, this is the center-point of a circle circumscribed around the triangle. If the triangle is obtuse, then this point is outside the triangle and if the triangle is a right triangle, then the point is the midpoint of the hypotenuse.
Incentre.
Circumcentre.
Centre of area of a triangle is the centre point for a circumscribed radius which will pass through all 3 vertices of the triangle.
The point of a triangle is called a vertex, same goes with rectangles and all other shapes with points.