Drawing a Venn diagram with three circles in a certain style

@schtandard already answered my question. However, for those who are looking for a drawing that matches the description, I provide the adapted version of @schtandard's solution:

begin{tikzpicture}
    deffirstcircle {(210:1.1cm) circle (1.5cm)}
    defsecondcircle{(330:1.1cm) circle (1.5cm)}
    defthirdcircle {( 90:1.1cm) circle (1.5cm)}
            
    colorlet{circle edge}{red!50}
    colorlet{circle area}{red!20}
            
    tikzset{filled/.style={fill=circle area, draw=circle edge, thick},
        outline/.style={draw=circle edge, thick}}
            
    setlength{parskip}{5mm}
            
    begin{scope}
        clip secondcircle;
        clip firstcircle;
        fill[filled] thirdcircle;
    end{scope}
            
    draw[outline] firstcircle node {$M_1$};
    draw[outline] secondcircle node {$M_2$};
    draw[outline] thirdcircle node {$M_3$};
            
    node[anchor=south] at (current bounding box.north) {$M_1 cap M_2 cap M_3$};
end{tikzpicture}

In this Asymptote solution we create a set of the key points first, then using them as the centers of the circles and corresponding begin-end points of the circular arcs, we construct an addressable list of the area boundaries, so each area can be filled differently.

The color used for the intersecting region is chosen as an average of the colors of corresponding intersecting regions.

The named points are shown for convenience.

//
// Venn-diag.asy
//
// run 
//   asy Venn-diag.asy
//
//  to get a standalone  Venn-diag.pdf
//
settings.tex="pdflatex";
size(7cm);
import fontsize;defaultpen(fontsize(8pt));
texpreamble("usepackage{lmodern}"+"usepackage{amsmath}"
+"usepackage{amsfonts}"+"usepackage{amssymb}");
pair A,B,C,D,E,F,O;
transform t=rotate(120);
O=0;   A=(0,-1/sqrt(3));
B=t*A; C=t*B;
D=A-1; E=A+1; F=2B-E;
guide[] g={arc(A,D,E)&arc(B,E,A,CW)&arc(C,A,D,CW)&cycle,};
g.cyclic=true;
g.push(t*g[-1]);
g.push(t*g[-1]);
g.push(arc(B,A,E)&arc(A,E,B)&arc(C,B,A,CW)&cycle);
g.push(t*g[-1]);
g.push(t*g[-1]);
g.push(arc(C,A,B)&arc(A,B,C)&arc(B,C,A)&cycle);
pen[] fPen={lightred,deepgreen,lightblue,};
fPen.push(0.5*(fPen[0]+fPen[1]));
fPen.push(0.5*(fPen[1]+fPen[2]));
fPen.push(0.5*(fPen[2]+fPen[0]));
fPen.push(1/3*(fPen[0]+fPen[1]+fPen[2]));
for(int i=0;i<fPen.length;++i) fill(g[i],fPen[i]);
draw(circle(A,1),red+.7*bp);      
draw(circle(B,1),deepgreen+.7*bp);
draw(circle(C,1),blue+.7*bp);     
string[] s=array("ABCDEF");
pair[] p={A,B,C,D,E,F,};
dot(p,UnFill);
for(int i=0;i<p.length;++i) label("$"+s[i]+"$",p[i],dir(p[i]));  
pair L=A-(0,1/2);
for(int i=0;i<3;++i)label("$mathbf M_"+string(i+1)+"$",t^i*L,white); 
L=(A+B+E)/3*1.2;
label("$M_1cap M_2$",L,white); 
label("$M_2cap M_3$",t*L,white); 
label("$M_1cap M_3$",t^2*L,white); 
label("$M_1cap M_2cap M_3$",white);