Imagine a Cartesian plane, where \(O \) is the origin at point \((0,0)\). Construct a triangle around this origin where the origin is the center of gravity of the triangle and is also the incenter of the triangle.
What properties must the triangle have? Please choose the most specific option. I.e. if any isosceles triangle works, choose "isosceles" but if only equilateral triangles work, choose equilateral triangles.
Assume that the triangle cannot be a line, i.e. if \(a,b\) and \(c\) are sides of a triangle, and \(a+b<c\).