Lucas Inner Triangle

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