Trying to learn.. what may be interesting geometric properties of associated center of a triangle (sides $a,b,c$) whose barycentric coordinates are:
$$\left(a^2+b^2-c^2, b^2+c^2-a^2, c^2+a^2-b^2 \right)$$
$$\left(ab-c^2, bc-c^2, ca-b^2 \right)$$
$$\left(\dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c}\right)$$
$$\left(\dfrac{1}{a^2},\dfrac{1}{b^2},\dfrac{1}{c^2}\right)$$
Appreciate pointer to any geometric software with three given barycentral functions as: $$f(a,b,c), g(a,b,c), h(a,b,c)\;$$ to locate recognize/ interpret/characterize the centre.