Mixtilinear incircles of a triangle

The mixtilinear incircle of a triangle is a special circle associated with a triangle, particularly in relation to its vertices and its incircle. For a given triangle \( ABC \), the mixtilinear incircle pertaining to a vertex, say \( A \), is the circle that is tangent to: 1. The incircle of triangle \( ABC \), 2. The arc \( BC \) of the circumcircle of triangle \( ABC \) that does not contain the vertex \( A \).