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
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.