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 $A$ is called the $A$ -mixtilinear incircle. Every triangle has three unique mixtilinear incircles, one corresponding to each vertex.

A

-Mixtilinear incircle of triangle

ABC

Proof of existence and uniqueness

The $A$ -excircle of triangle $ABC$ is unique. Let $\Phi$ be a transformation defined by the composition of an inversion centered at $A$ with radius ${\sqrt {AB\cdot AC}}$ and a reflection with respect to the angle bisector on $A$ . Since inversion and reflection are bijective and preserve touching points, then $\Phi$ does as well. Then, the image of the $A$ -excircle under $\Phi$ is a circle internally tangent to sides $AB,AC$ and the circumcircle of $ABC$ , that is, the $A$ -mixtilinear incircle. Therefore, the $A$ -mixtilinear incircle exists and is unique, and a similar argument can prove the same for the mixtilinear incircles corresponding to $B$ and $C$ .[1]

Construction

The hexagon

XCABYT_{A}

and the intersections

D,I,E

of its 3 pairs of opposite sides.

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

Draw the incenter $I$ by intersecting angle bisectors.
Draw a line through $I$ perpendicular to the line $AI$ , touching lines $AB$ and $AC$ at points $D$ and $E$ respectively. These are the tangent points of the mixtilinear circle.
Draw perpendiculars to $AB$ and $AC$ through points $D$ and $E$ respectively, and intersect them in $O_{A}$ . $O_{A}$ is the center of the circle, so a circle with center $O_{A}$ and radius $O_{A}E$ 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 $\Gamma$ be the circumcircle of triangle $ABC$ and $T_{A}$ be the tangency point of the $A$ -mixtilinear incircle $\Omega _{A}$ and $\Gamma$ . Let $X\neq T_{A}$ be the intersection of line $T_{A}D$ with $\Gamma$ and $Y\neq T_{A}$ be the intersection of line $T_{A}E$ with $\Gamma$ . Homothety with center on $T_{A}$ between $\Omega _{A}$ and $\Gamma$ implies that $X,Y$ are the midpoints of $\Gamma$ arcs $AB$ and $AC$ respectively. The inscribed angle theorem implies that $X,I,C$ and $Y,I,B$ are triples of collinear points. Pascal's theorem on hexagon $XCABYT_{A}$ inscribed in $\Gamma$ implies that $D,I,E$ are collinear. Since the angles $\angle {DAI}$ and $\angle {IAE}$ are equal, it follows that $I$ is the midpoint of segment $DE$ .[1]

Other properties

Radius

The following formula relates the radius $r$ of the incircle and the radius $\rho _{A}$ of the $A$ -mixtilinear incircle of a triangle $ABC$ :

r=\rho _{A}\cdot \cos ^{2}{\frac {\alpha }{2}}

where $\alpha$ is the magnitude of the angle at $A$ .[3]

Relationship with points on the circumcircle

The midpoint of the arc $BC$ that contains point $A$ is on the line $T_{A}I$ .[4][5]
The quadrilateral $T_{A}XAY$ is harmonic, which means that $T_{A}A$ is a symmedian on triangle $XT_{A}Y$ .[1]

Circles related to the tangency point with the circumcircle

$T_{A}BDI$ and $T_{A}CEI$ are cyclic quadrilaterals.[4]

Spiral similarities

$T_{A}$ is the center of a spiral similarity that maps $B,I$ to $I,C$ respectively.[1]

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.[3] The Online Encyclopedia of Triangle Centers lists this point as X(56).[6] It is defined by trilinear coordinates

{\frac {a}{c+a-b}}:{\frac {b}{c+a-b}}:{\frac {c}{a+b-c}},

and barycentric coordinates

{\frac {a^{2}}{c+a-b}}:{\frac {b^{2}}{c+a-b}}:{\frac {c^{2}}{a+b-c}}.

Radical center

The radical center of the three mixtilinear incircles is the point $J$ which divides $OI$ in the ratio

OJ:JI=2R:-r

where $I,r,O,R$ are the incenter, inradius, circumcenter and circumradius respectively.[5]

References

Baca, Jafet. "On Mixtilinear Incircles" (PDF). Retrieved October 27, 2021.
Weisstein, Eric W. "Mixtilinear Incircles". mathworld.wolfram.com. Retrieved 2021-10-31.
Yui, Paul (April 23, 2018). "Mixtilinear Incircles". The American Mathematical Monthly. 106 (10): 952–955. doi:10.1080/00029890.1999.12005146. Retrieved October 27, 2021.
Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. United States of America: MAA. p. 68. ISBN 978-1-61444-411-4.
Nguyen, Khoa Lu (2006). "On Mixtilinear Incircles and Excircles" (PDF). Retrieved November 27, 2021.
"ENCYCLOPEDIA OF TRIANGLE CENTERS". faculty.evansville.edu. Retrieved 2021-10-31.

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[:1-1] Baca, Jafet. "On Mixtilinear Incircles" (PDF). Retrieved October 27, 2021.

[2] Weisstein, Eric W. "Mixtilinear Incircles". mathworld.wolfram.com. Retrieved 2021-10-31.

[:0-3] Yui, Paul (April 23, 2018). "Mixtilinear Incircles". The American Mathematical Monthly. 106 (10): 952–955. doi:10.1080/00029890.1999.12005146. Retrieved October 27, 2021.

[:2-4] Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. United States of America: MAA. p. 68. ISBN 978-1-61444-411-4.

[:3-5] Nguyen, Khoa Lu (2006). "On Mixtilinear Incircles and Excircles" (PDF). Retrieved November 27, 2021.

[6] "ENCYCLOPEDIA OF TRIANGLE CENTERS". faculty.evansville.edu. Retrieved 2021-10-31.