Circular Triangle


A triangle ABC formed by three circular arcs. By extending the arcs into complete circles, the points of intersection A^', B^', and C^' are obtained. This gives the three circular triangles, A^'B^'C^', AB^'C^', A^'BC^', and A^'B^'C, which are called the associated triangles to ABC.


The circular triangle and its associated circles have a total of eight incircles and six circumcircles. These systems of circles have some remarkable properties, including the Hart circle, which is an analog of the nine-point circle in Feuerbach's theorem.

A closed-form set of formulas for the area of circular triangles like ABC is given by Fewell (2006).

See also

Apollonius' Problem, Arc, Associated Triangles, Circle-Circle Intersection, Feuerbach's Theorem, Hart Circle, Haruki's Theorem, Johnson's Theorem, Nine-Point Circle, Spherical Triangle

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Fewell, M. "Area of Common Overlap of Three Circles." Australian Dept. Defense. Oct. 2006., R. "Properties of a Circular Triangle." §397-404 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 251-257, 1893.

Referenced on Wolfram|Alpha

Circular Triangle

Cite this as:

Weisstein, Eric W. "Circular Triangle." From MathWorld--A Wolfram Web Resource.

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