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

Every triangle has three unique mixtilinear incircles, one corresponding to each vertex.

be a transformation defined by the composition of an inversion centered at

and a reflection with respect to the angle bisector on

Since inversion and reflection are bijective and preserve touching points, then

is a circle internally tangent to sides

{\displaystyle AB,AC}

-mixtilinear incircle exists and is unique, and a similar argument can prove the same for the mixtilinear incircles corresponding to

-mixtilinear incircle can be constructed with the following sequence of steps.

[2] This construction is possible because of the following fact: The incenter is the midpoint of the touching points of the mixtilinear incircle with the two sides.

be the circumcircle of triangle

be the tangency point of the

be the intersection of line

Homothety with center on

The inscribed angle theorem implies that

are triples of collinear points.

Pascal's theorem on hexagon

is the midpoint of segment

[1] The following formula relates the radius

is the magnitude of the angle at

are cyclic quadrilaterals.

is the center of a spiral similarity that maps

[1] 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.

[3] The Online Encyclopedia of Triangle Centers lists this point as X(56).

[6] It is defined by trilinear coordinates:

and barycentric coordinates:

The radical center of the three mixtilinear incircles is the point

are the incenter, inradius, circumcenter and circumradius respectively.