Unable to prove this question related to isogonal mappings The following question was asked in my Complex analysis quiz and I couldn't make much progress on the question.
So, I am posting it here for help.
A map is called isogonal when it preserves only angle size but not orierentation. For example $f(z)=\bar z$.

Problem: Prove that both $f(\bar z)$ and $\overline {f(z)} $  are both isogonal at points where f(z) is analytic with non zero derivative.

I am unable to proceed in this question. I took f(z) to analytic function with $f'(z)\neq 0$ but not able to understand  on how will I show that angles are preserved.( For f(z)=$\bar z$ it is easy as positive real axis is mapped to positive  real axis and positive imaginary axis is mapped to negative imaginary axis and similar impact will happen over others).
But, I am not sure how operating by f will have impact of $\bar z$ as  nature of f can be quite varied and how it will have impact on  overline of f(z).
Can you please guide me through that!
 A: This answer is an attempt to give a visual understanding. Apologies if it is too simplistic.
Imagine that $U$ is a $10\times40$ grid centered on some section of the real axis.  The figure below shows part of $f(U)$.  I'm assuming you already know that if $f$ is analytic it is also conformal, i.e. preserves angles.  The image of the real axis is shown as a thick blue line, and the rest of the grid is a mapping of the original grid.

The red and blue arrows above the line are images of two arrows.  $\bar{z}$ will flip the pre-images of the arrows across the real line, preserving angles, so the two arrows below the blue line are their image under $f(\bar{z}).$  Because $f$ preserves angles, the angles between the pairs of arrows are the same.
(actually, the image arrows will generally be curved line segments, and the angle that is preserved is measured at their intersection.)
Showing that $\overline{f(z)}$ is isogonal is easier, because it is just $f(z)$ flipped across the real axis.
A: Hint: If $f$ preserves angle size but reverses the orientation on some $U\subset\mathbb{C}$, then what does the function $g$ defined by $g(z) = f(\bar{z})$ do to angles on $U$? How do we characterise such a function complex analysis?
Conversely: if $f$ is analytic on $U$, what does it do to angles? What do the functions $g$ and $h$, defined by $g(z) = f(\bar{z})$ respectively $h(z)= \overline{f(z)}$ do to angles in this case?
