Tikz: The common tangent and the shaded region











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What are possible options to construct the tangent line (along with the shaded region) as shown below?



enter image description here



MWE:



documentclass[tikz, border=1cm]{standalone}
begin{document}
begin{tikzpicture}
coordinate (A) at (0,0);
coordinate (B) at (0,7);
coordinate (C) at (7,7);
coordinate (D) at (7,0);
coordinate (E) at (0,4);
coordinate (F) at (3,7);
coordinate (G) at (7,4);
coordinate (H) at (3,0);
coordinate (M) at (5,2);
coordinate (N) at (1.5,5.5);
draw (A)--(B)--(C)--(D)--cycle;
draw (E)--(G);
draw (F)--(H);
draw (N) circle [radius=1.5];
draw (M) circle [radius=2];
end{tikzpicture}
end{document}









share|improve this question




























    up vote
    10
    down vote

    favorite
    5












    What are possible options to construct the tangent line (along with the shaded region) as shown below?



    enter image description here



    MWE:



    documentclass[tikz, border=1cm]{standalone}
    begin{document}
    begin{tikzpicture}
    coordinate (A) at (0,0);
    coordinate (B) at (0,7);
    coordinate (C) at (7,7);
    coordinate (D) at (7,0);
    coordinate (E) at (0,4);
    coordinate (F) at (3,7);
    coordinate (G) at (7,4);
    coordinate (H) at (3,0);
    coordinate (M) at (5,2);
    coordinate (N) at (1.5,5.5);
    draw (A)--(B)--(C)--(D)--cycle;
    draw (E)--(G);
    draw (F)--(H);
    draw (N) circle [radius=1.5];
    draw (M) circle [radius=2];
    end{tikzpicture}
    end{document}









    share|improve this question


























      up vote
      10
      down vote

      favorite
      5









      up vote
      10
      down vote

      favorite
      5






      5





      What are possible options to construct the tangent line (along with the shaded region) as shown below?



      enter image description here



      MWE:



      documentclass[tikz, border=1cm]{standalone}
      begin{document}
      begin{tikzpicture}
      coordinate (A) at (0,0);
      coordinate (B) at (0,7);
      coordinate (C) at (7,7);
      coordinate (D) at (7,0);
      coordinate (E) at (0,4);
      coordinate (F) at (3,7);
      coordinate (G) at (7,4);
      coordinate (H) at (3,0);
      coordinate (M) at (5,2);
      coordinate (N) at (1.5,5.5);
      draw (A)--(B)--(C)--(D)--cycle;
      draw (E)--(G);
      draw (F)--(H);
      draw (N) circle [radius=1.5];
      draw (M) circle [radius=2];
      end{tikzpicture}
      end{document}









      share|improve this question















      What are possible options to construct the tangent line (along with the shaded region) as shown below?



      enter image description here



      MWE:



      documentclass[tikz, border=1cm]{standalone}
      begin{document}
      begin{tikzpicture}
      coordinate (A) at (0,0);
      coordinate (B) at (0,7);
      coordinate (C) at (7,7);
      coordinate (D) at (7,0);
      coordinate (E) at (0,4);
      coordinate (F) at (3,7);
      coordinate (G) at (7,4);
      coordinate (H) at (3,0);
      coordinate (M) at (5,2);
      coordinate (N) at (1.5,5.5);
      draw (A)--(B)--(C)--(D)--cycle;
      draw (E)--(G);
      draw (F)--(H);
      draw (N) circle [radius=1.5];
      draw (M) circle [radius=2];
      end{tikzpicture}
      end{document}






      tikz-pgf






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      edited yesterday

























      asked yesterday









      blackened

      1,366713




      1,366713






















          2 Answers
          2






          active

          oldest

          votes

















          up vote
          14
          down vote



          accepted










          Let me start by repeating the nice solution by LoopSpace, to whom I give full credit for the first part.



          documentclass[tikz, border=1cm]{standalone}
          usetikzlibrary{calc}
          begin{document}
          begin{tikzpicture}
          coordinate (A) at (0,0);
          coordinate (B) at (0,7);
          coordinate (C) at (7,7);
          coordinate (D) at (7,0);
          coordinate (E) at (0,4);
          coordinate (F) at (3,7);
          coordinate (G) at (7,4);
          coordinate (H) at (3,0);
          coordinate (M) at (5,2);
          coordinate (N) at (1.5,5.5);
          draw (A)--(B)--(C)--(D)--cycle;
          draw (E)--(G);
          draw (F)--(H);
          pgfmathsetmacro{rone}{1.5}
          pgfmathsetmacro{rtwo}{2}
          pgfmathsetmacro{mid}{rone/(rone + rtwo)}
          pgfmathsetmacro{out}{rone/(rone - rtwo)}
          node[circle,minimum size=2*rone*1cm,draw] (c1) at (N){};
          node[circle,minimum size=2*rtwo*1cm,draw] (c2) at (M){};
          path (c1.center) -- node[coordinate,pos=mid] (mid) {} (c2.center);
          path (c1.center) -- node[coordinate,pos=out] (out) {} (c2.center);
          foreach i in {1,2}
          {foreach j in {1,2}
          {foreach k in {mid,out}
          {coordinate (tijk) at (tangent cs:node=ci,point={(k)},solution=j);}}}
          foreach i in {2}
          {
          draw[red] ($(t1i out)!-1cm!(t2i out)$) -- ($(t2i out)!-1cm!(t1i out)$);
          }

          end{tikzpicture}
          end{document}


          enter image description here



          However, this setup is so simple that I cannot refrain from adding an analytic determination of the tangent. (The other possible tangents can be added completely analogously). The observations that go into the analytic determination are




          1. The slope of the tangent is given by the slope of the line connecting the centers of the circles plus the ratio of the difference of the radii and the distance of the centers.

          2. Given the slope, the respective points on the circle are uniquely determined (modulo 180).


          One thus arrives at



          documentclass[tikz, border=1cm]{standalone}
          usetikzlibrary{calc,backgrounds}
          begin{document}
          begin{tikzpicture}[tangent of circles/.style args={%
          at #1 and #2 with radii #3 and #4}{insert path={%
          let p1=($(#2)-(#1)$),n1={atan2(y1,x1)},n2={veclen(y1,x1)*1pt/1cm},
          n3={atan2(#4-#3,n2)}
          in ($(#1)+(n3+n1+90:#3)$) -- ($(#2)+(n3+n1+90:#4)$)}}]
          coordinate (A) at (0,0);
          coordinate (B) at (0,7);
          coordinate (C) at (7,7);
          coordinate (D) at (7,0);
          coordinate (E) at (0,4);
          coordinate (F) at (3,7);
          coordinate (G) at (7,4);
          coordinate (H) at (3,0);
          coordinate (M) at (5,2);
          coordinate (N) at (1.5,5.5);
          draw (A)--(B)--(C)--(D)--cycle;
          draw (E)--(G);
          draw (F)--(H);
          draw (N) circle [radius=1.5];
          draw (M) circle [radius=2];
          path[tangent of circles={at N and M with radii 1.5 and 2}]
          coordinate[pos=0] (aux0) coordinate[pos=1] (aux1);
          % extend the tangent
          draw (intersection cs:first line={(aux0)--(aux1)}, second line={(C)--(D)})
          -- (intersection cs:first line={(aux0)--(aux1)}, second line={(C)--(B)});
          % fill the region above right of the tangent
          begin{scope}[on background layer]
          fill[gray!50] (intersection cs:first line={(aux0)--(aux1)},
          second line={(E)--(G)}) -| (C) -|
          (intersection cs:first line={(aux0)--(aux1)}, second line={(F)--(H)})
          -- cycle;
          end{scope}
          % draw the little squares
          draw[fill=gray!20] (C) rectangle ++ (-0.4,-0.4)
          (F) rectangle ++ (0.4,-0.4)
          (G) rectangle ++ (-0.4,0.4)
          (intersection cs:first line={(E)--(G)}, second line={(F)--(H)})
          rectangle ++ (0.4,0.4);
          draw[fill,thick,-latex] (N) circle (1pt) -- ++(225:1.5) node[midway,above
          left]
          {$vec r_2$};
          draw[fill,thick,-latex] (M) circle (1pt) -- ++(225:2) node[midway,above
          left]
          {$vec r_1$};
          node at (barycentric cs:C=1,G=1,F=1) {$S_x$};
          end{tikzpicture}
          end{document}


          enter image description here



          Let me mention that I made no effort in shortening the code. One could kick out some coordinates, but I do not see any point in this. IMHO it would make the code just harder to understand.






          share|improve this answer























          • The remaining annotation may be added with node at (barycentric cs:C=1,G=1,F=1) {$S_x$};.
            – marmot
            yesterday






          • 1




            @blackened I changed it (and also moved the labels, as suggested by Artificial Stupidity). However, I do not add an animation, if you want an animation, see here, and wait for a PSTricks variant ;-)
            – marmot
            yesterday




















          up vote
          7
          down vote













          A PSTricks solution only for comparison purposes.



          documentclass[pstricks,border=12pt,12pt]{standalone}
          usepackage{pstricks-add,pst-eucl}
          begin{document}
          pspicture[PointName=none,PointSymbol=none](8,8)
          pnodes(4,0){A}(4,8){B}(0,4){C}(8,4){D}(2,6){P}(6,2){Q}
          psCircleTangents(P){2}(Q){2}
          pstInterLL{CircleTO1}{CircleTO2}{A}{B}{X}
          pstInterLL{CircleTO1}{CircleTO2}{C}{D}{Y}
          pspolygon*[linecolor=lightgray](X)(B)(D|B)(D)(Y)
          pcline[nodesep=-1.2](CircleTO1)(CircleTO2)
          psframe(D|B)
          psline(A)(B)
          psline(C)(D)
          pscircle(P){2}
          pscircle(Q){2}
          rput(A|D){psframe(12pt,12pt)}
          rput{90}(D){psframe(12pt,12pt)}
          rput{-90}(B){psframe(12pt,12pt)}
          rput{180}(D|B){psframe(12pt,12pt)}
          rput(Q|P){$S_x$}
          pcline{<-}([angle=225,nodesep=2]P)(P)naput{$r_2$}
          pcline{<-}([angle=225,nodesep=2]Q)(Q)naput{$r_1$}
          endpspicture
          end{document}


          enter image description here



          Different Radii



          documentclass[pstricks,border=12pt,12pt]{standalone}
          usepackage{pstricks-add,pst-eucl,pst-calculate}

          begin{document}
          foreach x in {4,4.5,...,6.0}{%
          pspicture[PointName=none,PointSymbol=none](8,8)
          pnodes(x,0){A}(A|0,8){B}(!0 8 xspace sub){C}(8,0|C){D}(!xspace 2 div dup neg 8 add){P}(!xspace 2 div dup 4 add exch neg 4 add){Q}
          psCircleTangents(P){pscalculate{x/2}}(Q){pscalculate{(8-x)/2}}
          pstInterLL{CircleTO1}{CircleTO2}{A}{B}{X}
          pstInterLL{CircleTO1}{CircleTO2}{C}{D}{Y}
          pspolygon*[linecolor=lightgray](X)(B)(D|B)(D)(Y)
          pcline[nodesep=-2](CircleTO1)(CircleTO2)
          psframe(D|B)
          psline(A)(B)
          psline(C)(D)
          pscircle(P){pscalculate{x/2}}
          pscircle(Q){pscalculate{(8-x)/2}}
          rput(A|D){psframe(12pt,12pt)}
          rput{90}(D){psframe(12pt,12pt)}
          rput{-90}(B){psframe(12pt,12pt)}
          rput{180}(D|B){psframe(12pt,12pt)}
          rput(Q|P){$S_x$}
          pcline{<-}([angle=225,nodesep=pscalculate{x/2}]P)(P)naput{$r_2$}
          pcline{<-}([angle=225,nodesep=pscalculate{(8-x)/2}]Q)(Q)naput{$r_1$}
          endpspicture}
          end{document}


          enter image description here






          share|improve this answer























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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            14
            down vote



            accepted










            Let me start by repeating the nice solution by LoopSpace, to whom I give full credit for the first part.



            documentclass[tikz, border=1cm]{standalone}
            usetikzlibrary{calc}
            begin{document}
            begin{tikzpicture}
            coordinate (A) at (0,0);
            coordinate (B) at (0,7);
            coordinate (C) at (7,7);
            coordinate (D) at (7,0);
            coordinate (E) at (0,4);
            coordinate (F) at (3,7);
            coordinate (G) at (7,4);
            coordinate (H) at (3,0);
            coordinate (M) at (5,2);
            coordinate (N) at (1.5,5.5);
            draw (A)--(B)--(C)--(D)--cycle;
            draw (E)--(G);
            draw (F)--(H);
            pgfmathsetmacro{rone}{1.5}
            pgfmathsetmacro{rtwo}{2}
            pgfmathsetmacro{mid}{rone/(rone + rtwo)}
            pgfmathsetmacro{out}{rone/(rone - rtwo)}
            node[circle,minimum size=2*rone*1cm,draw] (c1) at (N){};
            node[circle,minimum size=2*rtwo*1cm,draw] (c2) at (M){};
            path (c1.center) -- node[coordinate,pos=mid] (mid) {} (c2.center);
            path (c1.center) -- node[coordinate,pos=out] (out) {} (c2.center);
            foreach i in {1,2}
            {foreach j in {1,2}
            {foreach k in {mid,out}
            {coordinate (tijk) at (tangent cs:node=ci,point={(k)},solution=j);}}}
            foreach i in {2}
            {
            draw[red] ($(t1i out)!-1cm!(t2i out)$) -- ($(t2i out)!-1cm!(t1i out)$);
            }

            end{tikzpicture}
            end{document}


            enter image description here



            However, this setup is so simple that I cannot refrain from adding an analytic determination of the tangent. (The other possible tangents can be added completely analogously). The observations that go into the analytic determination are




            1. The slope of the tangent is given by the slope of the line connecting the centers of the circles plus the ratio of the difference of the radii and the distance of the centers.

            2. Given the slope, the respective points on the circle are uniquely determined (modulo 180).


            One thus arrives at



            documentclass[tikz, border=1cm]{standalone}
            usetikzlibrary{calc,backgrounds}
            begin{document}
            begin{tikzpicture}[tangent of circles/.style args={%
            at #1 and #2 with radii #3 and #4}{insert path={%
            let p1=($(#2)-(#1)$),n1={atan2(y1,x1)},n2={veclen(y1,x1)*1pt/1cm},
            n3={atan2(#4-#3,n2)}
            in ($(#1)+(n3+n1+90:#3)$) -- ($(#2)+(n3+n1+90:#4)$)}}]
            coordinate (A) at (0,0);
            coordinate (B) at (0,7);
            coordinate (C) at (7,7);
            coordinate (D) at (7,0);
            coordinate (E) at (0,4);
            coordinate (F) at (3,7);
            coordinate (G) at (7,4);
            coordinate (H) at (3,0);
            coordinate (M) at (5,2);
            coordinate (N) at (1.5,5.5);
            draw (A)--(B)--(C)--(D)--cycle;
            draw (E)--(G);
            draw (F)--(H);
            draw (N) circle [radius=1.5];
            draw (M) circle [radius=2];
            path[tangent of circles={at N and M with radii 1.5 and 2}]
            coordinate[pos=0] (aux0) coordinate[pos=1] (aux1);
            % extend the tangent
            draw (intersection cs:first line={(aux0)--(aux1)}, second line={(C)--(D)})
            -- (intersection cs:first line={(aux0)--(aux1)}, second line={(C)--(B)});
            % fill the region above right of the tangent
            begin{scope}[on background layer]
            fill[gray!50] (intersection cs:first line={(aux0)--(aux1)},
            second line={(E)--(G)}) -| (C) -|
            (intersection cs:first line={(aux0)--(aux1)}, second line={(F)--(H)})
            -- cycle;
            end{scope}
            % draw the little squares
            draw[fill=gray!20] (C) rectangle ++ (-0.4,-0.4)
            (F) rectangle ++ (0.4,-0.4)
            (G) rectangle ++ (-0.4,0.4)
            (intersection cs:first line={(E)--(G)}, second line={(F)--(H)})
            rectangle ++ (0.4,0.4);
            draw[fill,thick,-latex] (N) circle (1pt) -- ++(225:1.5) node[midway,above
            left]
            {$vec r_2$};
            draw[fill,thick,-latex] (M) circle (1pt) -- ++(225:2) node[midway,above
            left]
            {$vec r_1$};
            node at (barycentric cs:C=1,G=1,F=1) {$S_x$};
            end{tikzpicture}
            end{document}


            enter image description here



            Let me mention that I made no effort in shortening the code. One could kick out some coordinates, but I do not see any point in this. IMHO it would make the code just harder to understand.






            share|improve this answer























            • The remaining annotation may be added with node at (barycentric cs:C=1,G=1,F=1) {$S_x$};.
              – marmot
              yesterday






            • 1




              @blackened I changed it (and also moved the labels, as suggested by Artificial Stupidity). However, I do not add an animation, if you want an animation, see here, and wait for a PSTricks variant ;-)
              – marmot
              yesterday

















            up vote
            14
            down vote



            accepted










            Let me start by repeating the nice solution by LoopSpace, to whom I give full credit for the first part.



            documentclass[tikz, border=1cm]{standalone}
            usetikzlibrary{calc}
            begin{document}
            begin{tikzpicture}
            coordinate (A) at (0,0);
            coordinate (B) at (0,7);
            coordinate (C) at (7,7);
            coordinate (D) at (7,0);
            coordinate (E) at (0,4);
            coordinate (F) at (3,7);
            coordinate (G) at (7,4);
            coordinate (H) at (3,0);
            coordinate (M) at (5,2);
            coordinate (N) at (1.5,5.5);
            draw (A)--(B)--(C)--(D)--cycle;
            draw (E)--(G);
            draw (F)--(H);
            pgfmathsetmacro{rone}{1.5}
            pgfmathsetmacro{rtwo}{2}
            pgfmathsetmacro{mid}{rone/(rone + rtwo)}
            pgfmathsetmacro{out}{rone/(rone - rtwo)}
            node[circle,minimum size=2*rone*1cm,draw] (c1) at (N){};
            node[circle,minimum size=2*rtwo*1cm,draw] (c2) at (M){};
            path (c1.center) -- node[coordinate,pos=mid] (mid) {} (c2.center);
            path (c1.center) -- node[coordinate,pos=out] (out) {} (c2.center);
            foreach i in {1,2}
            {foreach j in {1,2}
            {foreach k in {mid,out}
            {coordinate (tijk) at (tangent cs:node=ci,point={(k)},solution=j);}}}
            foreach i in {2}
            {
            draw[red] ($(t1i out)!-1cm!(t2i out)$) -- ($(t2i out)!-1cm!(t1i out)$);
            }

            end{tikzpicture}
            end{document}


            enter image description here



            However, this setup is so simple that I cannot refrain from adding an analytic determination of the tangent. (The other possible tangents can be added completely analogously). The observations that go into the analytic determination are




            1. The slope of the tangent is given by the slope of the line connecting the centers of the circles plus the ratio of the difference of the radii and the distance of the centers.

            2. Given the slope, the respective points on the circle are uniquely determined (modulo 180).


            One thus arrives at



            documentclass[tikz, border=1cm]{standalone}
            usetikzlibrary{calc,backgrounds}
            begin{document}
            begin{tikzpicture}[tangent of circles/.style args={%
            at #1 and #2 with radii #3 and #4}{insert path={%
            let p1=($(#2)-(#1)$),n1={atan2(y1,x1)},n2={veclen(y1,x1)*1pt/1cm},
            n3={atan2(#4-#3,n2)}
            in ($(#1)+(n3+n1+90:#3)$) -- ($(#2)+(n3+n1+90:#4)$)}}]
            coordinate (A) at (0,0);
            coordinate (B) at (0,7);
            coordinate (C) at (7,7);
            coordinate (D) at (7,0);
            coordinate (E) at (0,4);
            coordinate (F) at (3,7);
            coordinate (G) at (7,4);
            coordinate (H) at (3,0);
            coordinate (M) at (5,2);
            coordinate (N) at (1.5,5.5);
            draw (A)--(B)--(C)--(D)--cycle;
            draw (E)--(G);
            draw (F)--(H);
            draw (N) circle [radius=1.5];
            draw (M) circle [radius=2];
            path[tangent of circles={at N and M with radii 1.5 and 2}]
            coordinate[pos=0] (aux0) coordinate[pos=1] (aux1);
            % extend the tangent
            draw (intersection cs:first line={(aux0)--(aux1)}, second line={(C)--(D)})
            -- (intersection cs:first line={(aux0)--(aux1)}, second line={(C)--(B)});
            % fill the region above right of the tangent
            begin{scope}[on background layer]
            fill[gray!50] (intersection cs:first line={(aux0)--(aux1)},
            second line={(E)--(G)}) -| (C) -|
            (intersection cs:first line={(aux0)--(aux1)}, second line={(F)--(H)})
            -- cycle;
            end{scope}
            % draw the little squares
            draw[fill=gray!20] (C) rectangle ++ (-0.4,-0.4)
            (F) rectangle ++ (0.4,-0.4)
            (G) rectangle ++ (-0.4,0.4)
            (intersection cs:first line={(E)--(G)}, second line={(F)--(H)})
            rectangle ++ (0.4,0.4);
            draw[fill,thick,-latex] (N) circle (1pt) -- ++(225:1.5) node[midway,above
            left]
            {$vec r_2$};
            draw[fill,thick,-latex] (M) circle (1pt) -- ++(225:2) node[midway,above
            left]
            {$vec r_1$};
            node at (barycentric cs:C=1,G=1,F=1) {$S_x$};
            end{tikzpicture}
            end{document}


            enter image description here



            Let me mention that I made no effort in shortening the code. One could kick out some coordinates, but I do not see any point in this. IMHO it would make the code just harder to understand.






            share|improve this answer























            • The remaining annotation may be added with node at (barycentric cs:C=1,G=1,F=1) {$S_x$};.
              – marmot
              yesterday






            • 1




              @blackened I changed it (and also moved the labels, as suggested by Artificial Stupidity). However, I do not add an animation, if you want an animation, see here, and wait for a PSTricks variant ;-)
              – marmot
              yesterday















            up vote
            14
            down vote



            accepted







            up vote
            14
            down vote



            accepted






            Let me start by repeating the nice solution by LoopSpace, to whom I give full credit for the first part.



            documentclass[tikz, border=1cm]{standalone}
            usetikzlibrary{calc}
            begin{document}
            begin{tikzpicture}
            coordinate (A) at (0,0);
            coordinate (B) at (0,7);
            coordinate (C) at (7,7);
            coordinate (D) at (7,0);
            coordinate (E) at (0,4);
            coordinate (F) at (3,7);
            coordinate (G) at (7,4);
            coordinate (H) at (3,0);
            coordinate (M) at (5,2);
            coordinate (N) at (1.5,5.5);
            draw (A)--(B)--(C)--(D)--cycle;
            draw (E)--(G);
            draw (F)--(H);
            pgfmathsetmacro{rone}{1.5}
            pgfmathsetmacro{rtwo}{2}
            pgfmathsetmacro{mid}{rone/(rone + rtwo)}
            pgfmathsetmacro{out}{rone/(rone - rtwo)}
            node[circle,minimum size=2*rone*1cm,draw] (c1) at (N){};
            node[circle,minimum size=2*rtwo*1cm,draw] (c2) at (M){};
            path (c1.center) -- node[coordinate,pos=mid] (mid) {} (c2.center);
            path (c1.center) -- node[coordinate,pos=out] (out) {} (c2.center);
            foreach i in {1,2}
            {foreach j in {1,2}
            {foreach k in {mid,out}
            {coordinate (tijk) at (tangent cs:node=ci,point={(k)},solution=j);}}}
            foreach i in {2}
            {
            draw[red] ($(t1i out)!-1cm!(t2i out)$) -- ($(t2i out)!-1cm!(t1i out)$);
            }

            end{tikzpicture}
            end{document}


            enter image description here



            However, this setup is so simple that I cannot refrain from adding an analytic determination of the tangent. (The other possible tangents can be added completely analogously). The observations that go into the analytic determination are




            1. The slope of the tangent is given by the slope of the line connecting the centers of the circles plus the ratio of the difference of the radii and the distance of the centers.

            2. Given the slope, the respective points on the circle are uniquely determined (modulo 180).


            One thus arrives at



            documentclass[tikz, border=1cm]{standalone}
            usetikzlibrary{calc,backgrounds}
            begin{document}
            begin{tikzpicture}[tangent of circles/.style args={%
            at #1 and #2 with radii #3 and #4}{insert path={%
            let p1=($(#2)-(#1)$),n1={atan2(y1,x1)},n2={veclen(y1,x1)*1pt/1cm},
            n3={atan2(#4-#3,n2)}
            in ($(#1)+(n3+n1+90:#3)$) -- ($(#2)+(n3+n1+90:#4)$)}}]
            coordinate (A) at (0,0);
            coordinate (B) at (0,7);
            coordinate (C) at (7,7);
            coordinate (D) at (7,0);
            coordinate (E) at (0,4);
            coordinate (F) at (3,7);
            coordinate (G) at (7,4);
            coordinate (H) at (3,0);
            coordinate (M) at (5,2);
            coordinate (N) at (1.5,5.5);
            draw (A)--(B)--(C)--(D)--cycle;
            draw (E)--(G);
            draw (F)--(H);
            draw (N) circle [radius=1.5];
            draw (M) circle [radius=2];
            path[tangent of circles={at N and M with radii 1.5 and 2}]
            coordinate[pos=0] (aux0) coordinate[pos=1] (aux1);
            % extend the tangent
            draw (intersection cs:first line={(aux0)--(aux1)}, second line={(C)--(D)})
            -- (intersection cs:first line={(aux0)--(aux1)}, second line={(C)--(B)});
            % fill the region above right of the tangent
            begin{scope}[on background layer]
            fill[gray!50] (intersection cs:first line={(aux0)--(aux1)},
            second line={(E)--(G)}) -| (C) -|
            (intersection cs:first line={(aux0)--(aux1)}, second line={(F)--(H)})
            -- cycle;
            end{scope}
            % draw the little squares
            draw[fill=gray!20] (C) rectangle ++ (-0.4,-0.4)
            (F) rectangle ++ (0.4,-0.4)
            (G) rectangle ++ (-0.4,0.4)
            (intersection cs:first line={(E)--(G)}, second line={(F)--(H)})
            rectangle ++ (0.4,0.4);
            draw[fill,thick,-latex] (N) circle (1pt) -- ++(225:1.5) node[midway,above
            left]
            {$vec r_2$};
            draw[fill,thick,-latex] (M) circle (1pt) -- ++(225:2) node[midway,above
            left]
            {$vec r_1$};
            node at (barycentric cs:C=1,G=1,F=1) {$S_x$};
            end{tikzpicture}
            end{document}


            enter image description here



            Let me mention that I made no effort in shortening the code. One could kick out some coordinates, but I do not see any point in this. IMHO it would make the code just harder to understand.






            share|improve this answer














            Let me start by repeating the nice solution by LoopSpace, to whom I give full credit for the first part.



            documentclass[tikz, border=1cm]{standalone}
            usetikzlibrary{calc}
            begin{document}
            begin{tikzpicture}
            coordinate (A) at (0,0);
            coordinate (B) at (0,7);
            coordinate (C) at (7,7);
            coordinate (D) at (7,0);
            coordinate (E) at (0,4);
            coordinate (F) at (3,7);
            coordinate (G) at (7,4);
            coordinate (H) at (3,0);
            coordinate (M) at (5,2);
            coordinate (N) at (1.5,5.5);
            draw (A)--(B)--(C)--(D)--cycle;
            draw (E)--(G);
            draw (F)--(H);
            pgfmathsetmacro{rone}{1.5}
            pgfmathsetmacro{rtwo}{2}
            pgfmathsetmacro{mid}{rone/(rone + rtwo)}
            pgfmathsetmacro{out}{rone/(rone - rtwo)}
            node[circle,minimum size=2*rone*1cm,draw] (c1) at (N){};
            node[circle,minimum size=2*rtwo*1cm,draw] (c2) at (M){};
            path (c1.center) -- node[coordinate,pos=mid] (mid) {} (c2.center);
            path (c1.center) -- node[coordinate,pos=out] (out) {} (c2.center);
            foreach i in {1,2}
            {foreach j in {1,2}
            {foreach k in {mid,out}
            {coordinate (tijk) at (tangent cs:node=ci,point={(k)},solution=j);}}}
            foreach i in {2}
            {
            draw[red] ($(t1i out)!-1cm!(t2i out)$) -- ($(t2i out)!-1cm!(t1i out)$);
            }

            end{tikzpicture}
            end{document}


            enter image description here



            However, this setup is so simple that I cannot refrain from adding an analytic determination of the tangent. (The other possible tangents can be added completely analogously). The observations that go into the analytic determination are




            1. The slope of the tangent is given by the slope of the line connecting the centers of the circles plus the ratio of the difference of the radii and the distance of the centers.

            2. Given the slope, the respective points on the circle are uniquely determined (modulo 180).


            One thus arrives at



            documentclass[tikz, border=1cm]{standalone}
            usetikzlibrary{calc,backgrounds}
            begin{document}
            begin{tikzpicture}[tangent of circles/.style args={%
            at #1 and #2 with radii #3 and #4}{insert path={%
            let p1=($(#2)-(#1)$),n1={atan2(y1,x1)},n2={veclen(y1,x1)*1pt/1cm},
            n3={atan2(#4-#3,n2)}
            in ($(#1)+(n3+n1+90:#3)$) -- ($(#2)+(n3+n1+90:#4)$)}}]
            coordinate (A) at (0,0);
            coordinate (B) at (0,7);
            coordinate (C) at (7,7);
            coordinate (D) at (7,0);
            coordinate (E) at (0,4);
            coordinate (F) at (3,7);
            coordinate (G) at (7,4);
            coordinate (H) at (3,0);
            coordinate (M) at (5,2);
            coordinate (N) at (1.5,5.5);
            draw (A)--(B)--(C)--(D)--cycle;
            draw (E)--(G);
            draw (F)--(H);
            draw (N) circle [radius=1.5];
            draw (M) circle [radius=2];
            path[tangent of circles={at N and M with radii 1.5 and 2}]
            coordinate[pos=0] (aux0) coordinate[pos=1] (aux1);
            % extend the tangent
            draw (intersection cs:first line={(aux0)--(aux1)}, second line={(C)--(D)})
            -- (intersection cs:first line={(aux0)--(aux1)}, second line={(C)--(B)});
            % fill the region above right of the tangent
            begin{scope}[on background layer]
            fill[gray!50] (intersection cs:first line={(aux0)--(aux1)},
            second line={(E)--(G)}) -| (C) -|
            (intersection cs:first line={(aux0)--(aux1)}, second line={(F)--(H)})
            -- cycle;
            end{scope}
            % draw the little squares
            draw[fill=gray!20] (C) rectangle ++ (-0.4,-0.4)
            (F) rectangle ++ (0.4,-0.4)
            (G) rectangle ++ (-0.4,0.4)
            (intersection cs:first line={(E)--(G)}, second line={(F)--(H)})
            rectangle ++ (0.4,0.4);
            draw[fill,thick,-latex] (N) circle (1pt) -- ++(225:1.5) node[midway,above
            left]
            {$vec r_2$};
            draw[fill,thick,-latex] (M) circle (1pt) -- ++(225:2) node[midway,above
            left]
            {$vec r_1$};
            node at (barycentric cs:C=1,G=1,F=1) {$S_x$};
            end{tikzpicture}
            end{document}


            enter image description here



            Let me mention that I made no effort in shortening the code. One could kick out some coordinates, but I do not see any point in this. IMHO it would make the code just harder to understand.







            share|improve this answer














            share|improve this answer



            share|improve this answer








            edited yesterday

























            answered yesterday









            marmot

            82.3k493176




            82.3k493176












            • The remaining annotation may be added with node at (barycentric cs:C=1,G=1,F=1) {$S_x$};.
              – marmot
              yesterday






            • 1




              @blackened I changed it (and also moved the labels, as suggested by Artificial Stupidity). However, I do not add an animation, if you want an animation, see here, and wait for a PSTricks variant ;-)
              – marmot
              yesterday




















            • The remaining annotation may be added with node at (barycentric cs:C=1,G=1,F=1) {$S_x$};.
              – marmot
              yesterday






            • 1




              @blackened I changed it (and also moved the labels, as suggested by Artificial Stupidity). However, I do not add an animation, if you want an animation, see here, and wait for a PSTricks variant ;-)
              – marmot
              yesterday


















            The remaining annotation may be added with node at (barycentric cs:C=1,G=1,F=1) {$S_x$};.
            – marmot
            yesterday




            The remaining annotation may be added with node at (barycentric cs:C=1,G=1,F=1) {$S_x$};.
            – marmot
            yesterday




            1




            1




            @blackened I changed it (and also moved the labels, as suggested by Artificial Stupidity). However, I do not add an animation, if you want an animation, see here, and wait for a PSTricks variant ;-)
            – marmot
            yesterday






            @blackened I changed it (and also moved the labels, as suggested by Artificial Stupidity). However, I do not add an animation, if you want an animation, see here, and wait for a PSTricks variant ;-)
            – marmot
            yesterday












            up vote
            7
            down vote













            A PSTricks solution only for comparison purposes.



            documentclass[pstricks,border=12pt,12pt]{standalone}
            usepackage{pstricks-add,pst-eucl}
            begin{document}
            pspicture[PointName=none,PointSymbol=none](8,8)
            pnodes(4,0){A}(4,8){B}(0,4){C}(8,4){D}(2,6){P}(6,2){Q}
            psCircleTangents(P){2}(Q){2}
            pstInterLL{CircleTO1}{CircleTO2}{A}{B}{X}
            pstInterLL{CircleTO1}{CircleTO2}{C}{D}{Y}
            pspolygon*[linecolor=lightgray](X)(B)(D|B)(D)(Y)
            pcline[nodesep=-1.2](CircleTO1)(CircleTO2)
            psframe(D|B)
            psline(A)(B)
            psline(C)(D)
            pscircle(P){2}
            pscircle(Q){2}
            rput(A|D){psframe(12pt,12pt)}
            rput{90}(D){psframe(12pt,12pt)}
            rput{-90}(B){psframe(12pt,12pt)}
            rput{180}(D|B){psframe(12pt,12pt)}
            rput(Q|P){$S_x$}
            pcline{<-}([angle=225,nodesep=2]P)(P)naput{$r_2$}
            pcline{<-}([angle=225,nodesep=2]Q)(Q)naput{$r_1$}
            endpspicture
            end{document}


            enter image description here



            Different Radii



            documentclass[pstricks,border=12pt,12pt]{standalone}
            usepackage{pstricks-add,pst-eucl,pst-calculate}

            begin{document}
            foreach x in {4,4.5,...,6.0}{%
            pspicture[PointName=none,PointSymbol=none](8,8)
            pnodes(x,0){A}(A|0,8){B}(!0 8 xspace sub){C}(8,0|C){D}(!xspace 2 div dup neg 8 add){P}(!xspace 2 div dup 4 add exch neg 4 add){Q}
            psCircleTangents(P){pscalculate{x/2}}(Q){pscalculate{(8-x)/2}}
            pstInterLL{CircleTO1}{CircleTO2}{A}{B}{X}
            pstInterLL{CircleTO1}{CircleTO2}{C}{D}{Y}
            pspolygon*[linecolor=lightgray](X)(B)(D|B)(D)(Y)
            pcline[nodesep=-2](CircleTO1)(CircleTO2)
            psframe(D|B)
            psline(A)(B)
            psline(C)(D)
            pscircle(P){pscalculate{x/2}}
            pscircle(Q){pscalculate{(8-x)/2}}
            rput(A|D){psframe(12pt,12pt)}
            rput{90}(D){psframe(12pt,12pt)}
            rput{-90}(B){psframe(12pt,12pt)}
            rput{180}(D|B){psframe(12pt,12pt)}
            rput(Q|P){$S_x$}
            pcline{<-}([angle=225,nodesep=pscalculate{x/2}]P)(P)naput{$r_2$}
            pcline{<-}([angle=225,nodesep=pscalculate{(8-x)/2}]Q)(Q)naput{$r_1$}
            endpspicture}
            end{document}


            enter image description here






            share|improve this answer



























              up vote
              7
              down vote













              A PSTricks solution only for comparison purposes.



              documentclass[pstricks,border=12pt,12pt]{standalone}
              usepackage{pstricks-add,pst-eucl}
              begin{document}
              pspicture[PointName=none,PointSymbol=none](8,8)
              pnodes(4,0){A}(4,8){B}(0,4){C}(8,4){D}(2,6){P}(6,2){Q}
              psCircleTangents(P){2}(Q){2}
              pstInterLL{CircleTO1}{CircleTO2}{A}{B}{X}
              pstInterLL{CircleTO1}{CircleTO2}{C}{D}{Y}
              pspolygon*[linecolor=lightgray](X)(B)(D|B)(D)(Y)
              pcline[nodesep=-1.2](CircleTO1)(CircleTO2)
              psframe(D|B)
              psline(A)(B)
              psline(C)(D)
              pscircle(P){2}
              pscircle(Q){2}
              rput(A|D){psframe(12pt,12pt)}
              rput{90}(D){psframe(12pt,12pt)}
              rput{-90}(B){psframe(12pt,12pt)}
              rput{180}(D|B){psframe(12pt,12pt)}
              rput(Q|P){$S_x$}
              pcline{<-}([angle=225,nodesep=2]P)(P)naput{$r_2$}
              pcline{<-}([angle=225,nodesep=2]Q)(Q)naput{$r_1$}
              endpspicture
              end{document}


              enter image description here



              Different Radii



              documentclass[pstricks,border=12pt,12pt]{standalone}
              usepackage{pstricks-add,pst-eucl,pst-calculate}

              begin{document}
              foreach x in {4,4.5,...,6.0}{%
              pspicture[PointName=none,PointSymbol=none](8,8)
              pnodes(x,0){A}(A|0,8){B}(!0 8 xspace sub){C}(8,0|C){D}(!xspace 2 div dup neg 8 add){P}(!xspace 2 div dup 4 add exch neg 4 add){Q}
              psCircleTangents(P){pscalculate{x/2}}(Q){pscalculate{(8-x)/2}}
              pstInterLL{CircleTO1}{CircleTO2}{A}{B}{X}
              pstInterLL{CircleTO1}{CircleTO2}{C}{D}{Y}
              pspolygon*[linecolor=lightgray](X)(B)(D|B)(D)(Y)
              pcline[nodesep=-2](CircleTO1)(CircleTO2)
              psframe(D|B)
              psline(A)(B)
              psline(C)(D)
              pscircle(P){pscalculate{x/2}}
              pscircle(Q){pscalculate{(8-x)/2}}
              rput(A|D){psframe(12pt,12pt)}
              rput{90}(D){psframe(12pt,12pt)}
              rput{-90}(B){psframe(12pt,12pt)}
              rput{180}(D|B){psframe(12pt,12pt)}
              rput(Q|P){$S_x$}
              pcline{<-}([angle=225,nodesep=pscalculate{x/2}]P)(P)naput{$r_2$}
              pcline{<-}([angle=225,nodesep=pscalculate{(8-x)/2}]Q)(Q)naput{$r_1$}
              endpspicture}
              end{document}


              enter image description here






              share|improve this answer

























                up vote
                7
                down vote










                up vote
                7
                down vote









                A PSTricks solution only for comparison purposes.



                documentclass[pstricks,border=12pt,12pt]{standalone}
                usepackage{pstricks-add,pst-eucl}
                begin{document}
                pspicture[PointName=none,PointSymbol=none](8,8)
                pnodes(4,0){A}(4,8){B}(0,4){C}(8,4){D}(2,6){P}(6,2){Q}
                psCircleTangents(P){2}(Q){2}
                pstInterLL{CircleTO1}{CircleTO2}{A}{B}{X}
                pstInterLL{CircleTO1}{CircleTO2}{C}{D}{Y}
                pspolygon*[linecolor=lightgray](X)(B)(D|B)(D)(Y)
                pcline[nodesep=-1.2](CircleTO1)(CircleTO2)
                psframe(D|B)
                psline(A)(B)
                psline(C)(D)
                pscircle(P){2}
                pscircle(Q){2}
                rput(A|D){psframe(12pt,12pt)}
                rput{90}(D){psframe(12pt,12pt)}
                rput{-90}(B){psframe(12pt,12pt)}
                rput{180}(D|B){psframe(12pt,12pt)}
                rput(Q|P){$S_x$}
                pcline{<-}([angle=225,nodesep=2]P)(P)naput{$r_2$}
                pcline{<-}([angle=225,nodesep=2]Q)(Q)naput{$r_1$}
                endpspicture
                end{document}


                enter image description here



                Different Radii



                documentclass[pstricks,border=12pt,12pt]{standalone}
                usepackage{pstricks-add,pst-eucl,pst-calculate}

                begin{document}
                foreach x in {4,4.5,...,6.0}{%
                pspicture[PointName=none,PointSymbol=none](8,8)
                pnodes(x,0){A}(A|0,8){B}(!0 8 xspace sub){C}(8,0|C){D}(!xspace 2 div dup neg 8 add){P}(!xspace 2 div dup 4 add exch neg 4 add){Q}
                psCircleTangents(P){pscalculate{x/2}}(Q){pscalculate{(8-x)/2}}
                pstInterLL{CircleTO1}{CircleTO2}{A}{B}{X}
                pstInterLL{CircleTO1}{CircleTO2}{C}{D}{Y}
                pspolygon*[linecolor=lightgray](X)(B)(D|B)(D)(Y)
                pcline[nodesep=-2](CircleTO1)(CircleTO2)
                psframe(D|B)
                psline(A)(B)
                psline(C)(D)
                pscircle(P){pscalculate{x/2}}
                pscircle(Q){pscalculate{(8-x)/2}}
                rput(A|D){psframe(12pt,12pt)}
                rput{90}(D){psframe(12pt,12pt)}
                rput{-90}(B){psframe(12pt,12pt)}
                rput{180}(D|B){psframe(12pt,12pt)}
                rput(Q|P){$S_x$}
                pcline{<-}([angle=225,nodesep=pscalculate{x/2}]P)(P)naput{$r_2$}
                pcline{<-}([angle=225,nodesep=pscalculate{(8-x)/2}]Q)(Q)naput{$r_1$}
                endpspicture}
                end{document}


                enter image description here






                share|improve this answer














                A PSTricks solution only for comparison purposes.



                documentclass[pstricks,border=12pt,12pt]{standalone}
                usepackage{pstricks-add,pst-eucl}
                begin{document}
                pspicture[PointName=none,PointSymbol=none](8,8)
                pnodes(4,0){A}(4,8){B}(0,4){C}(8,4){D}(2,6){P}(6,2){Q}
                psCircleTangents(P){2}(Q){2}
                pstInterLL{CircleTO1}{CircleTO2}{A}{B}{X}
                pstInterLL{CircleTO1}{CircleTO2}{C}{D}{Y}
                pspolygon*[linecolor=lightgray](X)(B)(D|B)(D)(Y)
                pcline[nodesep=-1.2](CircleTO1)(CircleTO2)
                psframe(D|B)
                psline(A)(B)
                psline(C)(D)
                pscircle(P){2}
                pscircle(Q){2}
                rput(A|D){psframe(12pt,12pt)}
                rput{90}(D){psframe(12pt,12pt)}
                rput{-90}(B){psframe(12pt,12pt)}
                rput{180}(D|B){psframe(12pt,12pt)}
                rput(Q|P){$S_x$}
                pcline{<-}([angle=225,nodesep=2]P)(P)naput{$r_2$}
                pcline{<-}([angle=225,nodesep=2]Q)(Q)naput{$r_1$}
                endpspicture
                end{document}


                enter image description here



                Different Radii



                documentclass[pstricks,border=12pt,12pt]{standalone}
                usepackage{pstricks-add,pst-eucl,pst-calculate}

                begin{document}
                foreach x in {4,4.5,...,6.0}{%
                pspicture[PointName=none,PointSymbol=none](8,8)
                pnodes(x,0){A}(A|0,8){B}(!0 8 xspace sub){C}(8,0|C){D}(!xspace 2 div dup neg 8 add){P}(!xspace 2 div dup 4 add exch neg 4 add){Q}
                psCircleTangents(P){pscalculate{x/2}}(Q){pscalculate{(8-x)/2}}
                pstInterLL{CircleTO1}{CircleTO2}{A}{B}{X}
                pstInterLL{CircleTO1}{CircleTO2}{C}{D}{Y}
                pspolygon*[linecolor=lightgray](X)(B)(D|B)(D)(Y)
                pcline[nodesep=-2](CircleTO1)(CircleTO2)
                psframe(D|B)
                psline(A)(B)
                psline(C)(D)
                pscircle(P){pscalculate{x/2}}
                pscircle(Q){pscalculate{(8-x)/2}}
                rput(A|D){psframe(12pt,12pt)}
                rput{90}(D){psframe(12pt,12pt)}
                rput{-90}(B){psframe(12pt,12pt)}
                rput{180}(D|B){psframe(12pt,12pt)}
                rput(Q|P){$S_x$}
                pcline{<-}([angle=225,nodesep=pscalculate{x/2}]P)(P)naput{$r_2$}
                pcline{<-}([angle=225,nodesep=pscalculate{(8-x)/2}]Q)(Q)naput{$r_1$}
                endpspicture}
                end{document}


                enter image description here







                share|improve this answer














                share|improve this answer



                share|improve this answer








                edited yesterday

























                answered yesterday









                Artificial Stupidity

                4,83111039




                4,83111039






























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