Hcfyk

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':

documentclassarticle
usepackagetikz
usetikzlibraryintersections,through
begindocument
begintikzpicture
 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'];


endtikzpicture
enddocument

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

documentclassarticle
usepackagetikz
usetikzlibraryintersections,through
makeatletter % from https://tex.stackexchange.com/a/127045/121799
tikzsetuse path/.code=tikz@addmodepgfsyssoftpath@setcurrentpath#1
makeatother
begindocument
begintikzpicture
 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) ;
 beginscope
 clip[use path=pathA];
 fill[blue,use path=pathB];
 endscope

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


endtikzpicture
enddocument

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

documentclassarticle
usepackagetikz
usetikzlibraryintersections,through,backgrounds
makeatletter % from https://tex.stackexchange.com/a/127045/121799
tikzsetuse path/.code=tikz@addmodepgfsyssoftpath@setcurrentpath#1
makeatother
begindocument
begintikzpicture
 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) ;
 beginscope[on background layer]
 clip[use path=pathA];
 fill[blue,use path=pathB];
 endscope

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


endtikzpicture
enddocument

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

documentclassarticle
usepackagetikz
usepackagepgfplots
usepgfplotslibraryfillbetween
pgfplotssetcompat=1.16
usetikzlibrarythrough
begindocument
begintikzpicture
 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];
endtikzpicture
enddocument

(Same output as above.)

The analytic determination of the arcs is another possibility.

documentclassarticle
usepackagetikz
usetikzlibraryintersections,through,calc
begindocument
begintikzpicture
 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);

endtikzpicture
enddocument