How to draw a circle (sphere) passing through four points?

Question

I am trying to draw a circle (sphere) passing through four points B, C, E, F like this picture

I tried with tikz-3dplot and my code

\documentclass[border=3mm,12pt]{standalone}
\usepackage{fouriernc}
\usepackage{tikz,tikz-3dplot} 
\usepackage{tkz-euclide}
\usetkzobj{all}

 \usetikzlibrary{intersections,calc,backgrounds}

 \begin{document}

 \tdplotsetmaincoords{70}{110}
  %\tdplotsetmaincoords{80}{100}
 \begin{tikzpicture}[tdplot_main_coords,scale=1.5]
 \pgfmathsetmacro\a{3}
 \pgfmathsetmacro\b{4}
 \pgfmathsetmacro\h{5}

 % definitions
 \path
 coordinate(A) at (0,0,0)
coordinate (B) at (\a,0,0)
coordinate (C) at (0,\b,0)                           
coordinate (S) at (0,0,\h)                
coordinate (E) at  ({\a*\h^2/(\a*\a + \h*\h)},0,{(\a*\a*\h)/(\a*\a + \h*\h)})
coordinate (F) at  (0,{(\b*\h*\h)/(\b*\b + \h*\h)},{(\b*\b*\h)/(\b*\b + \h*\h)});
 \draw[dashed,thick]
       (A) -- (B)  (A) -- (C)  (A) -- (E)  (S)--(A)  (F)--(A);
       \draw[thick]
       (S) -- (B) -- (C) -- cycle;
       \draw[thick] 
       (F) -- (B) (C)--(E) (F)--(E);
\tkzMarkRightAngle(S,E,A);
\tkzMarkRightAngle(S,F,A);

 \foreach \point/\position in {A/below,B/left,C/below,S/above,E/left,F/above}
 {
   \fill (\point) circle (.8pt);
   \node[\position=3pt] at (\point) {$\point$};
 }

 \end{tikzpicture}
 \end{document} 

and got

How can I draw a circle (sphere) passing through four points B, C, E, F?

Answer 1

A circle is determined by 3 points. A sphere, of course, needs at least 4 points on its boundary to be determined. However, the projection of the sphere, i.e. the circle, won't necessarily run through the projections of these points. (Actually, if the sphere is uniquely determined by these points, the boundary circle, i.e. the projection of the sphere on the screen coordinates, will never run through all projections of the points because for this to happen, the points need to lie in a plane, but then they no longer uniquely determine the circle.)

This shows two ways to construct circles that run through some of the points:

1. The dotted circle runs through F, E and C. It is fixed by this requirement. As a consequence it misses B by a small amount.
2. The red dashed circle runs through the midpoint of BC and through these points. It misses F and E by small amounts.

\documentclass[border=3mm,12pt]{standalone}
\usepackage{fouriernc}
\usepackage{tikz,tikz-3dplot} 
\usepackage{tkz-euclide}
\usetkzobj{all}
\usetikzlibrary{calc,through}
\tikzset{circle through 3 points/.style n args={3}{%
insert path={let    \p1=($(#1)!0.5!(#2)$),
                    \p2=($(#1)!0.5!(#3)$),
                    \p3=($(#1)!0.5!(#2)!1!-90:(#2)$),
                    \p4=($(#1)!0.5!(#3)!1!90:(#3)$),
                    \p5=(intersection of \p1--\p3 and \p2--\p4)
                    in },
at={(\p5)},
circle through= {(#1)}
}}

 \usetikzlibrary{intersections,calc,backgrounds}

 \begin{document}

 \tdplotsetmaincoords{70}{110}
  %\tdplotsetmaincoords{80}{100}
 \begin{tikzpicture}[tdplot_main_coords,scale=1.5]
 \pgfmathsetmacro\a{3}
 \pgfmathsetmacro\b{4}
 \pgfmathsetmacro\h{5}

 % definitions
 \path
 coordinate(A) at (0,0,0)
coordinate (B) at (\a,0,0)
coordinate (C) at (0,\b,0)                           
coordinate (S) at (0,0,\h)                
coordinate (E) at  ({\a*\h^2/(\a*\a + \h*\h)},0,{(\a*\a*\h)/(\a*\a + \h*\h)})
coordinate (F) at  (0,{(\b*\h*\h)/(\b*\b + \h*\h)},{(\b*\b*\h)/(\b*\b + \h*\h)});
 \draw[dashed,thick]
       (A) -- (B)  (A) -- (C)  (A) -- (E)  (S)--(A)  (F)--(A);
       \draw[thick]
       (S) -- (B) -- (C) -- cycle;
       \draw[thick] 
       (F) -- (B) (C)--(E) (F)--(E);
\tkzMarkRightAngle(S,E,A);
\tkzMarkRightAngle(S,F,A);

 \foreach \point/\position in {A/below,B/left,C/below,S/above,E/left,F/above}
 {
   \fill (\point) circle (.8pt);
   \node[\position=3pt] at (\point) {$\point$};
 }
  \node[circle through 3 points={F}{E}{C},draw=blue,dotted]{};
  \draw[red,dashed] 
  let \p1=($(B)-(C)$), \n1={veclen(\x1,\y1)/2} in ($(B)!0.5!(C)$) circle (\n1);
\end{tikzpicture}
\end{document}

It will be possible to construct the sphere as well. However, as mentioned its boundary may not run through any of the points.

ADDENDUM: In your setup, the four points do not determine a unique sphere because they all lie in a plane. Using Mathematica I was able to express F as a linear combination

 F = x B + y C + z E

where

So in this setup it is not possible to draw a unique sphere.

Answer 2

Thank you very much to marmot for correcting the circle BCEF here.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{calc,3dtools}
\begin{document}
    \begin{tikzpicture}[line cap=round,line join=round,
        3d/install view={phi=110,theta=70},
        declare function={a=3;b=4;h=5;},c/.style={circle,fill,inner sep=1pt}]
        \path
        (0,0,0) coordinate (A)  
        (a,0,0) coordinate (B)  
        (0,b,0) coordinate (C)                             
        (0,0,h)  coordinate (S)
        %(a/2,b/2,0)  coordinate (M)
        ;
        
        \path[3d/circumsphere center={A={(A)},B={(B)},C={(C)},D={(S)}}]
        coordinate (I);
        \pgfmathsetmacro{\myR}{sqrt(TD("(I)-(A)o(I)-(A)"))} ;
        \draw[3d/screen coords] (I) circle[radius=\myR];
        \path[3d/circumcircle center={A={(A)},B={(B)},C={(C)}}] coordinate (O);
        \path pic{3d/circle on sphere={R=\myR,C={(I)}, P={(O)}}};  
        \path[3d/line through={(S) and (B) named lSB}];
        \path[3d/line through={(S) and (C) named lSC}];
        \path[3d/project={(A) on lSB}] coordinate (E);
        \path[3d/project={(A) on lSC}] coordinate (F);
        
        %\path[3d/circumcircle center={A={(E)},B={(B)},C={(C)}}] coordinate (M);
        \path pic[draw=none]{3d circle through 3 points={%
                A={(B)},B={(E)},C={(F)},center name=M}};
        \pgfmathsetmacro{\myr}{tddistance("(B)","(M)")}%     
        \pgfmathsetmacro{\myMC}{TD("(C)-(M)")}
        \pgfmathtruncatemacro{\itest}{screendepth(\myMC)<0?0:1}
        \ifnum\itest=0
        \begin{scope}
            \clip[3d/screen coords] (I) circle[radius=\myR];
            \path pic[3d/hidden]{3d circle through 3 points={%
                    A={(B)},B={(E)},C={(F)},center name=M}};
        \end{scope}
        \begin{scope}
            \clip[3d/screen coords,even odd clip] (I) circle[radius=\myR]
            [generous outside path];
            \path pic[3d/visible]{3d circle through 3 points={%
                    A={(B)},B={(E)},C={(F)},center name=M}};
        \end{scope}
        \else
        \tikzset{3d/define orthonormal dreibein={A={(B)},B={(C)},C={(F)}}}
        \begin{scope}[x={(ex)},y={(ey)},z={(ez)},shift={(M)}]
            \draw[3d/hidden] (B) arc[start angle=180,end angle=0,radius=\myr];
            \draw[3d/visible] (B) arc[start angle=180,end angle=360,radius=\myr];
        \end{scope}
        \fi
        \path foreach \p/\g in {A/90,B/-90,C/0,S/90,E/180,F/90,I/0,M/-90}
        {(\p)node[c]{}+(\g:2.5mm) node{$\p$}};
        
        \draw[3d/hidden] (S) -- (A) (S) --(B) (S) -- (C) (A) -- (B) -- (C) -- cycle (A) -- (E) (A) -- (F);
        \end{tikzpicture}
\end{document}