Triangular diagram in Tikz

Question

I'm trying to replicate the following diagram in Tikz:

Where the borders encompass a right angled triangle, with dashed lines (not in picture) joining (0,0.5--1,0) and (0,1--0,5,0). Moreover, it has 2 dashed lines around the (1/3,1/3) point. These lines are kind of triangular, but smooth (so without kinks).

I have attempted the following:

\begin{tikzpicture}[scale=0.5]
    %Drawing the border
    \draw (0,0) -- (0,11) (0,0) -- (11,0);
    \draw [->] (0,11);
    \draw [->] (11,0);
    \draw (10,0) -- (0,10);
    % Drawing dashed lines
    \draw[dashed] (0,5) -- (10,0);
    \draw[dashed] (5,0) -- (0,10);
\end{tikzpicture}

But have no idea how to draw those dashed smooth triangles. Note that this need not to follow any particular equation, just a rough sketch is satisfactory :).

---

An astute commenter pointed out that the 1/3, 1/3 is on the intersection of the two lines! My drawn diagram forgot about this.

Answer 1

With smooth cycle you can draw smooth triangles. You can make it rounder by playing with tension.

\documentclass[tikz,border=3.14pt]{standalone}
\begin{document}
\begin{tikzpicture}[scale=0.5]
    %Drawing the border
    \draw (0,0) -- (0,11) (0,0) -- (11,0);
    \draw [->] (0,11);
    \draw [->] (11,0);
    \draw (10,0) -- (0,10);
    % Drawing dashed lines
    \draw[dashed] (0,5) -- (10,0);
    \draw[dashed] (5,0) -- (0,10);
    \pgfmathsetmacro{\myx}{2}
    \draw[dashed] plot[smooth cycle] coordinates {({10/3-\myx/3},{10/3+\myx})
    ({10/3+\myx},{10/3-\myx/3}) ({10/3-\myx/3},{10/3-\myx/3}) };
    \pgfmathsetmacro{\myx}{1}
    \draw[dashed] plot[smooth cycle] coordinates {({10/3-\myx/3},{10/3+\myx})
    ({10/3+\myx},{10/3-\myx/3}) ({10/3-\myx/3},{10/3-\myx/3}) };
\end{tikzpicture}
\end{document}

Answer 2

Same idea as @marmot, another code (varying the tension and the scale factor in the same time).

\documentclass[tikz,border=7pt]{standalone}
\begin{document}
  \begin{tikzpicture}
    %Drawing the border
    \draw (0,0) edge[-latex] (0,11) edge[-latex] (11,0) (10,0) -- (0,10);
    % Drawing dotted lines
    \draw[dotted, very thick] (0,5) -- (10,0) (5,0) -- (0,10);
    % The barycenter
    \fill[red] (10/3,10/3) coordinate (O) circle(3pt);
    % The smooth triangles
    \foreach[evaluate={\t=1-~/10}] ~ in {1,...,9}
        \draw[blue,scale around={~/10:(O)},smooth cycle,tension=\t,dashed, thick]
            plot coordinates {(0,0) (0,10) (10,0)};
  \end{tikzpicture}
\end{document}

Answer 3

An option that in my opinion is more practical, is to use the tkz-euclide package that is designed for this type of graphics, in the case of the drawing that you pose, you can use the centroid (or any other point) and look for points towards the vertices to then draw the internal triangles.

You can get this:

or This:

Here is the code:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% By J. Leon V.  coded based on the BSD, MIT, Beerware licences.
\documentclass[border=2mm]{standalone}
\usepackage{xcolor}
\usepackage{tkz-euclide}
\usetkzobj{all}

\begin{document}
    \begin{tikzpicture}
    % Set limits.
        \tkzInit[xmax=20,xmin=-1,ymax=20, ymin=-1]
    %\tkzGrid[sub,color=blue!10!,subxstep=.5,subystep=.5] %HIDE CARTESIAN GRID
    %\tkzAxeXY %HIDEN CARTESIAN AXIS
        \tkzClip
    %Define principal points. (10X)
    \tkzDefPoint(0,0){O}
    \tkzDefPoint(11,0){X1}
    \tkzDefPoint(0,11){X2}
     %Calculate points.
    \tkzDefBarycentricPoint(O=1,X1=10) \tkzGetPoint{A}
    \tkzDefBarycentricPoint(O=1,X2=10) \tkzGetPoint{B}  
    \tkzDefMidPoint(O,A) \tkzGetPoint{C}
    \tkzDefMidPoint(O,B) \tkzGetPoint{D}
    \tkzDefBarycentricPoint(O=2,A=1) \tkzGetPoint{c} 
    \tkzDefBarycentricPoint(O=2,B=1) \tkzGetPoint{d}
    \tkzCentroid(A,B,O) \tkzGetPoint{G} 

    %DRAW AND FIND POINTS
    \foreach \y [count=\i] in {1,..., 4}{
                \tkzDefBarycentricPoint(O=1,G=\y) \tkzGetPoint{GO\i} 
                \tkzDrawPoint[fill=red,size=10pt,](GO\i) % SHOW HIDEN POINTS GO
                \tkzDefBarycentricPoint(A=1,G=\y) \tkzGetPoint{GA\i} 
                \tkzDrawPoint[fill=red,size=10pt,](GA\i) % SHOW HIDEN POINTS GA
                \tkzDefBarycentricPoint(B=1,G=\y) \tkzGetPoint{GB\i} 
                \tkzDrawPoint[fill=red,size=10pt,](GB\i) % SHOW HIDEN POINTS GB
                \draw[dashed] plot[smooth cycle] coordinates{
                    (GO\i)(GA\i)(GB\i)
                    };
                }

    \tkzDrawSegments(B,A)
    \tkzDrawSegments[dashed](B,C A,D)
    \tkzDrawVectors[thick](O,X1 O,X2)
    \tkzMarkRightAngle(X2,O,X1) % For the case of 90 degres
    \tkzDrawPoints[size=10pt,shape=cross](c,d,G)

% 
%   %Labels:
    \tkzLabelPoint[below](A){ $A$}
    \tkzLabelPoint[below](C){$A/2$}
    \tkzLabelPoint[left](B){ $B$}
    \tkzLabelPoint[left](D){$B/2$}
    \tkzLabelPoint[left](O){$O$}
    \tkzLabelPoint[below](X1){$X_1$}
    \tkzLabelPoint[left](X2){$X_2$}
    \tkzLabelPoint[below](c){$A/3$}
    \tkzLabelPoint[left](d){$B/3$}
    \tkzLabelPoint[left](G){\sf Centroid}

    \end{tikzpicture}


\end{document}

Answer 4

You might also like a Metapost version, that shows how to do nice smooth cycle paths (in red). This is done using luamplib so compile with lualatex - or workout how to convert it for pdflatex with the gmp package (or plain MP).

\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\usepackage{luatex85}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
beginfig(1);
    numeric u;
    u = 144;
    path xx, yy;
    xx = origin -- right scaled 1.1u;
    yy = xx rotated 90;

    z0 = (1, 1) scaled 1/3 u;

    draw (0, 1/2u) -- (u, 0) -- (0, u) -- (1/2u, 0) 
         dashed evenly scaled 1/2 
         withcolor 1/2 white;

    dotlabel.llft("$x$", z0);

    % this is how to make a nice "smooth cycle" curve
    % default tension is 1 and gives something more like a circle
    % tension infinity gives straight lines
    def :: = .. tension 3 .. enddef;
    for t=1/5, 1/3:
        draw t[z0, (0,0)] ::
             t[z0, (0,u)] ::
             t[z0, (u,0)] :: cycle 
             dashed withdots scaled 1/4
             withcolor 2/3 red;
    endfor

    drawarrow xx; label.rt ("$x_1$", point 1 of xx);
    drawarrow yy; label.top("$x_2$", point 1 of yy);

    draw (down--up) scaled 1 shifted (1/3u,0);
    draw (down--up) scaled 1 shifted (1/2u,0);
    draw (down--up) scaled 1 shifted (   u,0);
    label.bot("$\frac13$", (1/3u, 0));
    label.bot("$\frac12$", (1/2u, 0));
    label.bot("$1$", (u,0));

    draw (left--right) scaled 1 shifted (0, 1/3u);
    draw (left--right) scaled 1 shifted (0, 1/2u);
    draw (left--right) scaled 1 shifted (0,    u);
    label.lft("$\frac13$", (0, 1/3u));
    label.lft("$\frac12$", (0, 1/2u));
    label.lft("$1$", (0, u));

endfig;
\end{mplibcode}
\end{document}