section_path: Miscellaneous triangles/Circular triangles
section_name: Circular triangles
paragraph_text: Circular triangles
  
  A circular triangle is a triangle with circular arc edges. The edges of a circular triangle may be either convex (bending outward) or concave (bending inward). The intersection of three disks forms a circular triangle whose sides are all convex. An example of a circular triangle with three convex edges is a Reuleaux triangle, which can be made by intersecting three circles of equal size. (The construction may be performed with a compass alone without needing a straightedge, by the Mohr–Mascheroni theorem.) Alternatively, it can be constructed by rounding the sides of an equilateral triangle.
  
  A special case of concave circular triangle can be seen in a pseudotriangle. A subset of a plane is convex if, given any two points in that subset, the whole line segment joining them also lies within that subset. A pseudotriangle is a simply-connected subset of the plane lying between three mutually tangent convex regions. These sides are three smoothed curved lines connecting their endpoints called the cusp points. Any pseudotriangle can be partitioned into many pseudotriangles with the boundaries of convex disks and bitangent lines, a process known as pseudo-triangulation. For n {\displaystyle n} disks in a pseudotriangle, the partition gives 2 n − 2 {\displaystyle 2n-2} pseudotriangles and 3 n − 3 {\displaystyle 3n-3} bitangent lines. The convex hull of any pseudotriangle is a triangle.