Basic Examples (2) 

Find the three mixtilinear incircles for a reference triangle:

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An equivalent specification:

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Scope (4) 

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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Applications (3) 

In the left most 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 w.r.t 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 X₅₆ (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 applies to B and C:

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Properties and Relations (1) 

Display all available information from MixtilinearIncircles in a dataset:

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Neat Examples (3) 

For a given triangle the point X₅₆ (blue), mixtilinear radical center (red), incenter (orange) and circumcenter (black) are collinear:

Show the collinearity:

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Visualize the points and the line passing the four points:

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