Mixtilinear incircles of a triangle
In plane geometry, a mixtilinear incircle of a triangle is a circle which is tangent to two of its sides and internally tangent to its circumcircle. The mixtilinear incircle of a triangle tangent to the two sides containing vertex  is called the -mixtilinear incircle. Every triangle has three unique mixtilinear incircles, one corresponding to each vertex. Proof of existence and uniqueness 

The -excircle of triangle  is unique. Let  be a transformation defined by the composition of an inversion centered at  with radius  and a reflection with respect to the angle bisector on . Since inversion and reflection are bijective and preserve touching points, then  does as well. Then, the image of the -excircle under  is a circle internally tangent to sides  and the circumcircle of , that is, the -mixtilinear incircle. Therefore, the -mixtilinear incircle exists and is unique, and a similar argument can prove the same for the mixtilinear incircles corresponding to  and . Construction 

The -mixtilinear incircle can be constructed with the following sequence of steps. Draw the incenter  by intersecting angle bisectors. Draw a line through  perpendicular to the line , touching lines  and  at points  and  respectively. These are the tangent points of the mixtilinear circle. Draw perpendiculars to  and  through points  and  respectively, and intersect them in . is the center of the circle, so a circle with center  and radius   is the mixtilinear incircle

This construction is possible because of the following fact:

Lemma 
The incenter is the midpoint of the touching points of the mixtilinear incircle with the two sides. Proof 
Let  be the circumcircle of triangle  and  be the tangency point of the -mixtilinear incircle  and . Let  be the intersection of line  with  and  be the intersection of line  with . Homothety with center on  between  and  implies that  are the midpoints of  arcs  and  respectively. The inscribed angle theorem implies that  and   are triples of collinear points. Pascal's theorem on hexagon  inscribed in  implies that  are collinear. Since the angles  and  are equal, it follows that  is the midpoint of segment . Other properties

Radius 
The following formula relates the radius  of the incircle and the radius  of the -mixtilinear incircle of a triangle :
 where  is the magnitude of the angle at . Relationship with points on the circumcircle 

 The midpoint of the arc  that contains point  is on the line . The quadrilateral  is harmonic, which means that  is a symmedian on triangle . Circles related to the tangency point with the circumcircle 

 and  are cyclic quadrilaterals. Spiral similarities 

 is the center of a spiral similarity that maps  to  respectively. Relationship between the three mixtilinear incircles

Lines joining vertices and mixtilinear tangency points 
The three lines joining a vertex to the point of contact of the circumcircle with the corresponding mixtilinear incircle meet at the external center of similitude of the incircle and circumcircle. The Online Encyclopedia of Triangle Centers lists this point as X(56). It is defined by trilinear coordinates 
 
and barycentric coordinates

Radical center 
The radical center of the three mixtilinear incircles is the point  which divides  in the ratio where  are the incenter, inradius, circumcenter and circumradius respectively. References 

Euclidean plane geometry