Basic Examples (2) 

Find the three mixtilinear incircles for a reference triangle:

In[1]:=

Out[1]=

An equivalent specification:

In[2]:=

Out[2]=

Scope (4) 

Find a list of properties for a triangle:

In[3]:=

Out[3]=

Visualize the three mixtilinear circles along with the reference triangle:

In[4]:=

Out[4]=

MixtilinearIncircles works with Triangle objects:

In[5]:=

Out[5]=

Find the mixtilinear incircles for several triangles:

In[6]:=

Out[6]=

Applications (3) 

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:

In[7]:=

In[8]:=

Out[8]=

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 X₅₆ (blue). The latter is the concurrence of the lines joining the vertex and the corresponding mixtilinear incircle-to-circumcircle contact point:

In[9]:=

Out[9]=

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

In[10]:=

Out[10]=

Properties and Relations (1) 

Display all available information from MixtilinearIncircles in a dataset:

In[11]:=

Out[11]=

Neat Examples (3) 

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

In[12]:=

Show the collinearity:

In[13]:=

Out[13]=

In[14]:=

Out[14]=

Visualize the points and the line passing the four points:

In[15]:=

Out[15]=