Misplaced Pages

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

Proof of existence and uniqueness

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 ${\displaystyle {\sqrt {AB\cdot AC}}}$ 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 ${\displaystyle C}$.

Construction

The ${\displaystyle A}$-mixtilinear incircle can be constructed with the following sequence of steps.

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

The incenter is the midpoint of the touching points of the mixtilinear incircle with the two sides.

Proof

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 ${\displaystyle X\neq T_{A}}$ be the intersection of line ${\displaystyle T_{A}D}$ with ${\displaystyle \Gamma }$ and ${\displaystyle Y\neq T_{A}}$ 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 ${\displaystyle DE}$.

Other properties

Radius

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 ${\displaystyle ABC}$:$${\displaystyle r=\rho _{A}\cdot \cos ^{2}{\frac {\alpha }{2}}}$$

where ${\displaystyle \alpha }$ is the magnitude of the angle at ${\displaystyle A}$.

Relationship with points on the circumcircle

• The midpoint of the arc ${\displaystyle BC}$ that contains point ${\displaystyle A}$ is on the line ${\displaystyle T_{A}I}$.
• The quadrilateral ${\displaystyle T_{A}XAY}$ is harmonic, which means that ${\displaystyle T_{A}A}$ is a symmedian on triangle ${\displaystyle XT_{A}Y}$.

Circles related to the tangency point with the circumcircle

${\displaystyle T_{A}BDI}$ and ${\displaystyle T_{A}CEI}$ are cyclic quadrilaterals.

Spiral similarities

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

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. The Online Encyclopedia of Triangle Centers lists this point as X(56). 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: $${\displaystyle {\frac {a^{2}}{b+c-a}}:{\frac {b^{2}}{c+a-b}}:{\frac {c^{2}}{a+b-c}}.}$$

Radical center

The radical center of the three mixtilinear incircles is the point ${\displaystyle J}$ which divides ${\displaystyle OI}$ in the ratio: $${\displaystyle OJ:JI=2R:-r}$$where ${\displaystyle I,r,O,R}$ are the incenter, inradius, circumcenter and circumradius respectively.

References

1. ^ Baca, Jafet. "On Mixtilinear Incircles" (PDF). Retrieved October 27, 2021.
2. Weisstein, Eric W. "Mixtilinear Incircles". mathworld.wolfram.com. Retrieved 2021-10-31.
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.
4. ^ Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. United States of America: MAA. p. 68. ISBN 978-1-61444-411-4.
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.

• Euclidean plane geometry