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I am trying to draw like this picture

I tried

documentclass[border=2 mm,12pt]{standalone}

usepackage{fouriernc}

usepackage{tikz}

usepackage{tikz-3dplot}

begin{document}

    tdplotsetmaincoords{70}{180}

    begin{tikzpicture}[tdplot_main_coords,line join = round, line cap = round]

        coordinate (A) at (0,0,0);

        coordinate (S) at (7/2,{-7*sqrt(3)/6},14/3);

        coordinate (C) at (7,0,0);

        coordinate (B) at  ({65/14},{15*sqrt(3)/14},0);

        coordinate (I) at  (7/2,{-7*sqrt(3)/6},0) ;



        draw[very thick] (A) -- (B) (B) -- (C)  (S) --  (A) (S) --(B) (S) -- (C) ;



        draw[dashed ] (C) -- (A) (I) -- (A) (I) -- (B) (I) --(C) (S) --(I) ;

        draw[very thick] (I) circle ({7/sqrt(3)});

        foreach point/position in {A/below,B/below,C/below,

            I/below,S/above}

        {

            fill (point) circle (1.5pt);

            node[position=3pt] at (point) {$point$};

        }

          end{tikzpicture}

  end{document}

and got

How to draw dashed arc of a circle behind pyramid?

One thing that always works is the reverseclip trick.

documentclass[border=2 mm,12pt]{standalone}

usepackage{fouriernc}

usepackage{tikz}

usepackage{tikz-3dplot}

% based on https://tex.stackexchange.com/a/12033/121799

tikzset{reverseclip/.style={insert path={(current bounding box.south west)rectangle 

  (current bounding box.north east)} }}

begin{document}

    tdplotsetmaincoords{70}{180}

    begin{tikzpicture}[tdplot_main_coords,line join = round, line cap = round]

        coordinate (A) at (0,0,0);

        coordinate (S) at (7/2,{-7*sqrt(3)/6},14/3);

        coordinate (C) at (7,0,0);

        coordinate (B) at  ({65/14},{15*sqrt(3)/14},0);

        coordinate (I) at  (7/2,{-7*sqrt(3)/6},0) ;



        draw[very thick] (A) -- (B) (B) -- (C)  (S) --  (A) (S) --(B) (S) -- (C) ;



        draw[dashed ] (C) -- (A) (I) -- (A) (I) -- (B) (I) --(C) (S) --(I) ;

        path (I) circle ({1.01*7/sqrt(3)});

        begin{scope}

         clip (S) -- (C) -- (B) -- (A) -- cycle [reverseclip];

         draw[very thick] (I) circle ({7/sqrt(3)});

        end{scope}

        begin{scope}

         clip (S) -- (C) -- (B) -- (A);

         draw[dashed] (I) circle ({7/sqrt(3)});

        end{scope}

        foreach point/position in {A/below,B/below,C/below,

            I/below,S/above}

        {

            fill (point) circle (1.5pt);

            node[position=3pt] at (point) {$point$};

        }

    end{tikzpicture}

end{document}

Using the intersections library, the code would look like this (I have highlighted in red the required line):

documentclass[border=2 mm,12pt]{standalone}

usepackage{fouriernc}

usepackage{tikz}

usepackage{tikz-3dplot}

usetikzlibrary{intersections}

colorlet{bgcolor}{white}



tikzset{

    overdraw/.style={preaction={draw,bgcolor,line width=#1}},

    overdraw/.default=2pt

}



begin{document}

    tdplotsetmaincoords{70}{180}

    begin{tikzpicture}[tdplot_main_coords,line join = round, line cap = round]

        coordinate (A) at (0,0,0);

        coordinate (S) at (7/2,{-7*sqrt(3)/6},14/3);

        coordinate (C) at (7,0,0);

        coordinate (B) at  ({65/14},{15*sqrt(3)/14},0);

        coordinate (I) at  (7/2,{-7*sqrt(3)/6},0) ;



        draw[very thick] (A) -- (B) (B) -- (C) (S) --(B);

        draw[very thick,name path=SC] (S) -- (C);

        draw[very thick,name path=SA] (S) --  (A);



        draw[very thick,name path=CIR] (I) circle ({7/sqrt(3)});



        path [name intersections={of=SC and CIR, by={C,C'}}];

        path [name intersections={of=SA and CIR, by={D,D'}}];

        draw[red,dashed,overdraw] (C') to [bend right=-10] (D'); %to draw the curved path

        draw[dashed] (C) -- (A) (I) -- (A) (I) -- (B) (I) --(C) (S) --(I) ;

        foreach point/position in {A/below,B/below,C/below,

            I/below,S/above}

        {

            fill (point) circle (1.5pt);

            node[position=3pt] at (point) {$point$};

        }

          end{tikzpicture}

  end{document}