The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex. (Wikipedia)
\documentclass[border=10pt]{standalone}
\usepackage[inline]{asymptote}
\begin{document}
\begin{asy}[width=\the\linewidth,inline=true]
path triangle = scale(1/2)*polygon(3);
pair a = point(triangle, 0), b=point(triangle,1), c=point(triangle,2);
path releaux = arc(c=a, b, c) & arc(c=b, c, a) & arc(c=c, a, b) & cycle;
fill(releaux, gray);
draw(releaux);
\end{asy}
\end{document}
end{asy}
end{document}
Source: TeX.SE
Author: Charles Staats (License)