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).
See also Apollonius' Problem
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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 London: Macmillian, pp. 251-257,
Elementary Treatise on Modern Pure Geometry. Referenced on Wolfram|Alpha Circular Triangle
Cite this as:
Weisstein, Eric W. "Circular Triangle."
From --A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/CircularTriangle.html