$T$ is a mapping, $S$ is a subset of domain of $T$, we say that $S$ is fixed by $T$ if $T(S)\subseteq S$.
(If $T$ is a linear mapping, $S$ is called an invariant subspace.)
The question is how to find 1-dimensional fixed sets (fixed curves) of a plane transformation.
Example. The translation $F(x,y)=(x+1,y)$ fixes all lines parallel to $x$ axis.
Fixed curves of $F$ are of the form $y=\text{const}$
Example. The rotation $F(x,y)=\left(\frac{\sqrt{3} x}{2}-\frac{y}{2},\frac{x}{2}+\frac{\sqrt{3} y}{2}\right)$ fixes all circles centered at origin.
Fixed curve of $F$ are of the form $x^2+y^2=\text{const}$
Example.
The centroid of the triangle with vertices $(0,0),(1,0),(x,y)$ is $F(x,y)=\left(\frac{x+1}{3},\frac{y}{3}\right)$
$F$ fixes all lines through $(\frac12,0)$.
So fixed curves of $F$ are of the form $\frac{x-\frac12}{y}=\text{const}$
Example.
The incenter of the triangle with vertices $(0,0),(1,0),(x,y)$ is $F(x,y)=\left(\frac{1}{2} \left(\sqrt{x^2+y^2}-\sqrt{(x-1)^2+y^2}+1\right),\frac{y}{\sqrt{x^2+y^2}+\sqrt{(x-1)^2+y^2}+1}\right)$
Note that the value of $\arctan\frac{y}{x},\arctan\frac{y}{1-x}$ will be halved after applying $F$. So their ratio stay constant.
So fixed curves of $F$ are of the form $\frac{\arctan\frac{y}{x}}{\arctan\frac{y}{1-x}}=\text{const}$
If the constant is irrational, this curve is transcendental; if the constant is rational, this curve is an algebraic curve. For example $\frac{\arctan\frac{y}{x}}{\arctan\frac{y}{1-x}}=2$ is a branch of a hyperbola.
Example.
$F(x,y)$ is X(1138) of the triangle with vertices $(0,0),(1,0),(x,y)$.
This post asks what the fixed curves of $F$ are.
Example.
The symmedian point of the triangle with vertices $(0,0),(1,0),(x,y)$ is $F(x,y)=\left(\frac{x^2+x+y^2}{2 \left((x-1) x+y^2+1\right)},\frac{y}{2 \left((x-1) x+y^2+1\right)}\right)$
This post asks what the fixed curves of $F$ are.
Question.
The nine-point center of the triangle with vertices $(0,0),(1,0),(x,y)$ is $F(x,y)=\left(\frac{1}{4} (2 x+1),\frac{-x^2+x+y^2}{4 y}\right)$
What are the fixed curves of $F$? (An obvious one is the line $x=\frac12$.)
