Taiji Functions of a Circle

Call a function a Taiji function of a circle if its graph divides both the area and the perimeter (circumference) of the circle into two equal parts. Determine, with proof, which of the following claims are true.

1. For any circle, its Taiji function is not unique. 2. If a function is a Taiji function of two circles, then the two circles are concentric. 3. The function $f(x)=x^3-3x^2+3x$ is a Taiji function of the circle $(x-1)^2+(y-1)^2=4$. 4. Every Taiji function of a circle is centrally symmetric.