# Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

Get the three mixtilinear incircles of a triangle

Contributed by:
Shenghui Yang

ResourceFunction["MixtilinearIncircles"][{ gives the three mixtilinear incircles for a triangle with vertices | |

ResourceFunction["MixtilinearIncircles"][{ returns additional related geometric objects of the mixtilinear incircles for the given | |

ResourceFunction["MixtilinearIncircles"][{ returns a list of properties. |

The mixtilinear incircles of a triangle are the three circles that are tangent to two sides and internally tangent to the circumcircle.

Given triangle *ABC*, the convention of the order for mixtilinear incenters and circles follows the diagram given here (points and circles are sorted with respect to alphabetic order in subscript):

Given triangle *ABC*, the mixtilinear radical axis for vertex *A* is the line through the intersection of circles *O*_{B} and *O*_{C} in the diagram. Those for vertices *B* and *C* are defined similarly.

Given triangle *ABC*, the mixtilinear circle passes through points *O*_{A}, *O*_{B} and *O*_{C}.

The following values are available for *property*:

"Circles" | the mixtilinear incircles of a triangle |

"MixtilinearIncircles" | same as "Circles" |

"MixtilinearIncenters" | the centers of mixtilinear incircles of a triangle |

"MixtilinearInradii" | the radii of mixtilinear incircles of a triangle |

"MixtilinearTriangle" | the triangle connecting the three mixtilinear incenters |

"MixtilinearCircle" | the circumcircle of the mixtilinear triangle |

"MixtilinearRadicalAxes" | the radical axes of pairs of the mixtilinear incircles |

"MixtilinearRadicalCenter" | the intersection of the radical axes |

"ContactsOnCirumcircle" | the contacts between mixtilinear incircles and circumcircle |

"OverlappingInversions" | combination of circle inversion and reflection along the angle bisector |

"X56" | the concurrence of lines joining vertices and contacts on the circumcircle |

All | an association including all properties given above |

ResourceFunction["MixtilinearIncircles"][Triangle[{*p*_{1},*p*_{2},*p*_{3}}]] is equivalent to ResourceFunction["MixtilinearIncircles"][{*p*_{1},*p*_{2},*p*_{3}}].

Find the three mixtilinear incircles for a reference triangle:

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An equivalent specification:

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Find a list of properties for a triangle:

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Visualize the three mixtilinear circles along with the reference triangle:

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MixtilinearIncircles works with Triangle objects:

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Find the mixtilinear incircles for several triangles:

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In the leftmost diagram, the overlapping inversion for vertex *A*={0,0} uses circle inversion at *A* with power and then applies the reflection transformation about the angular bisector of *A*. This operation sends the mixtilinear incircle and circumcircle contact point (black) to the blue point, which is the contact point between *A*-excircle and side *BC*:

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Therefore, the lines connecting *A* to the blue point and *A* to the black point are isogonal conjugate with respect to the angle bisector of *A*. Similar results apply to *B* and *C*.

The isogonal conjugacy also relates the Nagel point (brown) and the Kimberling center *X*_{56} (blue). The latter is the concurrence of the lines joining the vertex and the corresponding mixtilinear incircle-to-circumcircle contact point:

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Given triangle *ABC*, the overlapping inversion at *A* sends the incenter to the *A*-excenter. Similar results apply to *B* and *C*:

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Display all available information from MixtilinearIncircles in a dataset:

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For a given triangle, the point *X*_{56} (blue), mixtilinear radical center (red), incenter (orange) and circumcenter (black) are collinear:

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Show the collinearity:

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Visualize the points and the line passing the four points:

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- 1.0.1 – 22 April 2022
- 1.0.0 – 01 November 2021

This work is licensed under a Creative Commons Attribution 4.0 International License