Mixtilinear incircles of a triangle

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

${displaystyle A}$

is called the

${displaystyle A}$

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

Proof of existence and uniqueness[edit]

The

${displaystyle A}$

-excircle of triangle

${displaystyle ABC}$

is unique. Let

${displaystyle Phi }$

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

${displaystyle A}$

with radius

${displaystyle {sqrt {ABcdot AC}}}$

and a reflection with respect to the angle bisector on

${displaystyle A}$

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

${displaystyle Phi }$

does as well. Then, the image of the

${displaystyle A}$

-excircle under

${displaystyle Phi }$

is a circle internally tangent to sides

${displaystyle AB,AC}$

and the circumcircle of

${displaystyle ABC}$

, that is, the

${displaystyle A}$

-mixtilinear incircle. Therefore, the

${displaystyle A}$

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

${displaystyle B}$

and

${displaystyle C}$

.^[1]

Construction[edit]

The

${displaystyle A}$

-mixtilinear incircle can be constructed with the following sequence of steps.^[2]

1. Draw the incenter
   ${displaystyle I}$

   by intersecting angle bisectors.

2. Draw a line through
   ${displaystyle I}$

   perpendicular to the line

   ${displaystyle AI}$

   , touching lines

   ${displaystyle AB}$

   and

   ${displaystyle AC}$

   at points

   ${displaystyle D}$

   and

   ${displaystyle E}$

   respectively. These are the tangent points of the mixtilinear circle.

3. Draw perpendiculars to
   ${displaystyle AB}$

   and

   ${displaystyle AC}$

   through points

   ${displaystyle D}$

   and

   ${displaystyle E}$

   respectively, and intersect them in

   ${displaystyle O_{A}}$

   .

   ${displaystyle O_{A}}$

   is the center of the circle, so a circle with center

   ${displaystyle O_{A}}$

   and radius

   ${displaystyle O_{A}E}$

   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

${displaystyle Gamma }$

be the circumcircle of triangle

${displaystyle ABC}$

and

${displaystyle T_{A}}$

be the tangency point of the

${displaystyle A}$

-mixtilinear incircle

${displaystyle Omega _{A}}$

and

${displaystyle Gamma }$

. Let

${displaystyle Xneq T_{A}}$

be the intersection of line

${displaystyle T_{A}D}$

with

${displaystyle Gamma }$

and

${displaystyle Yneq T_{A}}$

be the intersection of line

${displaystyle T_{A}E}$

with

${displaystyle Gamma }$

. Homothety with center on

${displaystyle T_{A}}$

between

${displaystyle Omega _{A}}$

and

${displaystyle Gamma }$

implies that

${displaystyle X,Y}$

are the midpoints of

${displaystyle Gamma }$

arcs

${displaystyle AB}$

and

${displaystyle AC}$

respectively. The inscribed angle theorem implies that

${displaystyle X,I,C}$

and

${displaystyle Y,I,B}$

are triples of collinear points. Pascal’s theorem on hexagon

${displaystyle XCABYT_{A}}$

inscribed in

${displaystyle Gamma }$

implies that

${displaystyle D,I,E}$

are collinear. Since the angles

${displaystyle angle {DAI}}$

and

${displaystyle angle {IAE}}$

are equal, it follows that

${displaystyle I}$

is the midpoint of segment

${displaystyle DE}$

.^[1]

Other properties[edit]

Radius[edit]

The following formula relates the radius

${displaystyle r}$

of the incircle and the radius

${displaystyle rho _{A}}$

of the

${displaystyle A}$

-mixtilinear incircle of a triangle

${displaystyle ABC}$

:

where

${displaystyle alpha }$

is the magnitude of the angle at

${displaystyle A}$

.^[3]

Relationship with points on the circumcircle[edit]

Circles related to the tangency point with the circumcircle[edit]

${displaystyle T_{A}BDI}$

and

${displaystyle T_{A}CEI}$

are cyclic quadrilaterals.^[4]

Spiral similarities[edit]

${displaystyle T_{A}}$

is the center of a spiral similarity that maps

${displaystyle B,I}$

to

${displaystyle I,C}$

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

${displaystyle {frac {a}{c+a-b}}:{frac {b}{c+a-b}}:{frac {c}{a+b-c}}}$

and barycentric coordinates

${displaystyle {frac {a^{2}}{c+a-b}}:{frac {b^{2}}{c+a-b}}:{frac {c^{2}}{a+b-c}}}$

.

Radical center[edit]

The radical center of the three mixtilinear incircles is the point

${displaystyle J}$

which divides

${displaystyle OI}$

in the ratio

where

${displaystyle I,r,O,R}$

are the incenter, inradius, circumcenter and circumradius respectively.^[5]

References[edit]