All triangles are bicentric, i.e., possess both an incircle and a circumcircle. This is not necessarily the case for polygons with four or more sides.
The inradius 
and circumradius 
of a triangle are connected by
where 
is the distance between the incenter and circumcenter
(Coolidge 1971, p. 45).
See also
Bicentric Polygon, Bicentric Quadrilateral, Euler Triangle Formula
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References
Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, 1971.van Lamoen, F. M. "Bicentric Triangles." Nieuw Arch. Wisk. 17, 363-372, 1999.
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Cite this as:
Weisstein, Eric W. "Bicentric Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BicentricTriangle.html