Circular triangle

In

Intersection of three disks

Circular horn triangle

Reuleaux triangle

Arbelos

The intersection of three circular disks forms a convex circular triangle. For instance, a Reuleaux triangle is a special case of this construction where the three disks are centered on the vertices of an equilateral triangle, with radius equal to the side length of the triangle. However, not every convex circular triangle is formed as an intersection of disks in this way.

A circular horn triangle has all

A cardioid-like circular triangle found by Roger Joseph Boscovich has three vertices equally spaced on a line, two equal semicircles on one side of the line, and a third semicircle of twice the radius on the other side of the line. The two outer vertices have the interior angle ${\displaystyle \pi }$ and the middle vertex has interior angle ${\displaystyle 2\pi }$. It has the curious property that all lines through the middle vertex bisect its perimeter.^[3]

Other circular triangles can have a mixture of convex and concave circular arc edges.

Three given angles ${\displaystyle \theta _{1}}$, ${\displaystyle \theta _{2}}$, and ${\displaystyle \theta _{3}}$ in the interval ${\displaystyle [0,2\pi ]}$ form the interior angles of a circular triangle (without self-intersections) if and only if they obey the system of inequalities $${\displaystyle {\begin{aligned}-2\pi &<\theta _{1}+\theta _{2}-\theta _{3}<2\pi \\-2\pi &<\theta _{1}+\theta _{3}-\theta _{2}<2\pi \\-2\pi &<\theta _{2}+\theta _{3}-\theta _{1}<2\pi .\end{aligned}}}$$ All circular triangles with the same interior angles as each other are equivalent to each other under Möbius transformations.^[4]

Circular triangles give the solution to an

• Hart circle, a circle associated with certain circular triangles
• Hyperbolic triangle, a triangle that has straight sides in hyperbolic geometry, but is drawn as circular in some models of hyperbolic geometry
• Lune and Lens
  , two-sided figures bounded by circular arcs
• Sine-triple-angle circle
• Trefoil, a circular triangle bulging outward from its three vertices, used in architecture