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# Circular Triangle

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

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 is given by Fewell (2006).

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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## References

Fewell, M. "Area of Common Overlap of Three Circles." Australian Dept. Defense. Oct. 2006. http://www.dsto.defence.gov.au/publications/4815/DSTO-TN-0722.pdf.Lachlan, 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. https://mathworld.wolfram.com/CircularTriangle.html