How to fill the area between n intersecting points in TikZ

Question

Given two shapes, which are intersecting, how can we fill the area common to them in TikZ (i.e. the area between their intersecting points).

Here, the even odd rule might not work as it is not alternate filling but intersecting points.

Please note to determine the intersecting points, I am using the intersections library.

In the below MWE, how do I fill the area between C and C':

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections,through}
\begin{document}
\begin{tikzpicture}
  \coordinate (A) at (0,0);
  \coordinate (B) at (1.25,0.25);


  \node (E) [name path=E,draw,circle through=(B)] at (A) {};
  \node (F) [name path=F,draw,circle through=(A)] at (B) {};

  \path [name intersections={of=E and F, by={[label=above:$C$]C,[label=below:$C'$]C'}}];


\end{tikzpicture}
\end{document}

Answer 1

The common area is obtained by clipping against one of the circles and filling the other.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections,through}
\makeatletter % from https://tex.stackexchange.com/a/127045/121799
\tikzset{use path/.code=\tikz@addmode{\pgfsyssoftpath@setcurrentpath#1}}
\makeatother
\begin{document}
\begin{tikzpicture}
  \coordinate (A) at (0,0);
  \coordinate (B) at (1.25,0.25);


  \node (E) [name path=E,draw,circle through=(B),save path=\pathA] at (A) {};
  \node (F) [name path=F,draw,circle through=(A),save path=\pathB] at (B) {};
  \begin{scope}
  \clip[use path=\pathA];
  \fill[blue,use path=\pathB];
  \end{scope}

  \path [name intersections={of=E and F, by={[label=above:$C$]C,[label=below:$C'$]C'}}];


\end{tikzpicture}
\end{document}

In case you are concerned by the fact that the circle contour gets partly overpainted, use backgrounds.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections,through,backgrounds}
\makeatletter % from https://tex.stackexchange.com/a/127045/121799
\tikzset{use path/.code=\tikz@addmode{\pgfsyssoftpath@setcurrentpath#1}}
\makeatother
\begin{document}
\begin{tikzpicture}
  \coordinate (A) at (0,0);
  \coordinate (B) at (1.25,0.25);


  \node (E) [name path=E,draw,circle through=(B),save path=\pathA] at (A) {};
  \node (F) [name path=F,draw,circle through=(A),save path=\pathB] at (B) {};
  \begin{scope}[on background layer]
  \clip[use path=\pathA];
  \fill[blue,use path=\pathB];
  \end{scope}

  \path [name intersections={of=E and F, by={[label=above:$C$]C,[label=below:$C'$]C'}}];


\end{tikzpicture}
\end{document}

You can always fill intersection segments. (You can combine this with the background stuff above.)

\documentclass{article}
\usepackage{tikz}
\usepackage{pgfplots}
\usepgfplotslibrary{fillbetween}
\pgfplotsset{compat=1.16}
\usetikzlibrary{through}
\begin{document}
\begin{tikzpicture}
  \coordinate (A) at (0,0);
  \coordinate (B) at (1.25,0.25);


  \node (E) [name path=E,draw,circle through=(B)] at (A) {};
  \node (F) [name path=F,draw,circle through=(A)] at (B) {};

  \path [name intersections={of=E and F, by={[label=above:$C$]C,[label=below:$C'$]C'}}];

  \path[%draw,red,thick,
  fill=blue,
  intersection segments={of=E and F,sequence={L1--R2--L3}}];
\end{tikzpicture}
\end{document}

(Same output as above.)

The analytic determination of the arcs is another possibility.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections,through,calc}
\begin{document}
\begin{tikzpicture}
  \coordinate (A) at (0,0);
  \coordinate (B) at (1.25,0.25);


  \node (E) [name path=E,draw,circle through=(B)] at (A) {};
  \node (F) [name path=F,draw,circle through=(A)] at (B) {};

  \path [name intersections={of=E and F, by={[label=above:$C$]C,[label=below:$C'$]C'}}];
  \path[fill=blue] let \p1=($(A.center)-(B.center)$),\p2=($(C.center)-(A.center)$),
  \p3=($(C'.center)-(A.center)$),\p4=($(C.center)-(B.center)$),
  \p5=($(C'.center)-(B.center)$),
  \n1={veclen(\x2,\y2)}, % radius A
  \n2={veclen(\x4,\y4)}, % radius B
  \n3={atan2(\y2,\x2)}, % angle A 1
  \n4={atan2(\y3,\x3)}, % angle A 2
  \n5={atan2(\y4,\x4)}, % angle B 1
  \n6={atan2(\y5,\x5)} % angle B 2
  in (C) arc(\n3:\n4:\n1)  arc(\n6:\n5-360:\n2);

\end{tikzpicture}
\end{document}