In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an incircle). This is a circle that is tangent to each of the polygon's sides. The dual polygon of a tangential polygon is a cyclic polygon, which has a circumscribed circle passing through each of its vertices.
There exists a tangential polygon of n sequential sides a1, ..., an if and only if the system of equations
has a solution (x1, ..., xn) in positive reals. If such a solution exists, then x1, ..., xn are the tangent lengths of the polygon (the lengths from the vertices to the points where the incircle is tangent to the sides).
While all triangles are tangential to some circle, a triangle is called the tangential triangle of a reference triangle if the tangencies of the tangential triangle with the circle are also the vertices of the reference triangle.