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In the classical presentation of a three-set Venn diagram as three overlapping circles, the central region (representing elements belonging to all three sets) takes the shape of a Reuleaux triangle. The same three circles form one of the standard drawings of the Borromean rings, three mutually linked rings that cannot, however, be realized as geometric circles. Parts of these same circles are used to form the triquetra, a figure of three overlapping semicircles (each two of which form a vesica piscis symbol) that again has a Reuleaux triangle at its center; just as the three circles of the Venn diagram may be interlaced to form the Borromean rings, the three circular arcs of the triquetra may be interlaced to form a trefoil knot.

Relatives of the Reuleaux triangle arise in the problem of finding the minimum perimeter shape that encloses a fixed amount of area and includes three specified points in the plane. For a wide range of choices of the area parameter, the optimal solution to this problem will be a curved triangle whose three sides are circular arcs with equal radii. In particular, when the three points are equidistant from each other and the area is that of the Reuleaux triangle, the Reuleaux triangle is the optimal enclosure.

Circular triangles are triangles with circular-arc edges, including the Reuleaux triangle as well as other shapes.The deltoid curve is another type of curvilinear triangle, but one in which the curves replacing each side of an equilateral triangle are concave rather than convex. It is not composed of circular arcs, but may be formed by rolling one circle within another of three times the radius. Other planar shapes with three curved sides include the arbelos, which is formed from three semicircles with collinear endpoints, and the Bézier triangle.

The Reuleaux triangle may also be interpreted as the stereographic projection of one triangular face of a spherical tetrahedron, the Schwarz triangle of parameters 3 2, 3 2, 3 2 {\displaystyle {\tfrac {3}{2}},{\tfrac {3}{2}},{\tfrac {3}{2}}} with spherical angles of measure 120 ∘ {\displaystyle 120^{\circ }} and sides of spherical length arccos (− 1 3). {\displaystyle {\arccos }{\bigl (}{-{\tfrac {1}{3}}}{\bigr)}.}