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

-Mixtilinear incircle of triangle

Proof of existence and uniqueness

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

The hexagon and the intersections of its 3 pairs of opposite sides.

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

  1. Draw the incenter by intersecting angle bisectors.
  2. Draw a line through perpendicular to the line , touching lines and at points and respectively. These are the tangent points of the mixtilinear circle.
  3. 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

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

Proof

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

Radius

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

  • The midpoint of the arc that contains point is on the line .[4][5]
  • The quadrilateral is harmonic, which means that is a symmedian on triangle .[1]

and are cyclic quadrilaterals.[4]

Spiral similarities

is the center of a spiral similarity that maps to 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

and barycentric coordinates

Radical center

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

  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.
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