The points of tangency of the Lucas inner circle with the Lucas circles are the inverses of the vertices , , and in the Lucas circles radical circle. These form the Lucas inner triangle , a term coined here for the first time.
The Lucas inner triangle has trilinear vertex matrix
where , , and is Conway triangle notation (P. Moses, pers. comm., Jan. 13, 2005).
The area is
where is the area of the reference triangle.
The following table gives the centers of the Lucas inner triangle in terms of the centers of the reference triangle that correspond to Kimberling centers .
center of Lucas inner triangle | center of reference triangle | ||
first isodynamic point | first isodynamic point | ||
second isodynamic point | second isodynamic point | ||
Schoute center | Schoute center | ||
center of the Parry circle | center of the Parry circle | ||
isogonal conjugate of | isogonal conjugate of | ||
isogonal conjugate of | isogonal conjugate of |
The following table summarized the points at which the Lucas inner triangle is perspective with various other triangles.
triangle | center function of perspector |
Lucas central triangle | |
Lucas tangents triangle | |
reference triangle | |
symmedial triangle |