Mixtilinear incircles of a triangle
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In geometry, a mixtilinear incircle of a triangle is a circle 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[edit]
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
.[1]
Construction[edit]
The
-mixtilinear incircle can be constructed with the following sequence of steps.[2]
- 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[edit]
The incenter is the midpoint of the touching points of the mixtilinear incircle with the two sides.
Proof[edit]
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
.[1]
Other properties[edit]
Radius[edit]
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
.[3]
Relationship with points on the circumcircle[edit]
and
are cyclic quadrilaterals.[4]
Spiral similarities[edit]
is the center of a spiral similarity that maps
to
respectively.[1]
Relationship between the three mixtilinear incircles[edit]
Lines joining vertices and mixtilinear tangency points[edit]
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
.
Radical center[edit]
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.[5]
References[edit]
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