Ttgjyuk

I am trying to draw like this picture

I tried

documentclass[border=2 mm,12pt]standalone
usepackagefouriernc
usepackagetikz
usepackagetikz-3dplot
begindocument
 tdplotsetmaincoords70180
 begintikzpicture[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) $point$;
 
 endtikzpicture
 enddocument

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
usepackagefouriernc
usepackagetikz
usepackagetikz-3dplot
% based on https://tex.stackexchange.com/a/12033/121799
tikzsetreverseclip/.style=insert path=(current bounding box.south west)rectangle 
 (current bounding box.north east) 
begindocument
 tdplotsetmaincoords70180
 begintikzpicture[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));
 beginscope
 clip (S) -- (C) -- (B) -- (A) -- cycle [reverseclip];
 draw[very thick] (I) circle (7/sqrt(3));
 endscope
 beginscope
 clip (S) -- (C) -- (B) -- (A);
 draw[dashed] (I) circle (7/sqrt(3));
 endscope
 foreach point/position in A/below,B/below,C/below,
 I/below,S/above
 
 fill (point) circle (1.5pt);
 node[position=3pt] at (point) $point$;
 
 endtikzpicture
enddocument

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

documentclass[border=2 mm,12pt]standalone
usepackagefouriernc
usepackagetikz
usepackagetikz-3dplot
usetikzlibraryintersections
colorletbgcolorwhite

tikzset
 overdraw/.style=preaction=draw,bgcolor,line width=#1,
 overdraw/.default=2pt


begindocument
 tdplotsetmaincoords70180
 begintikzpicture[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) $point$;
 
 endtikzpicture
 enddocument