Consider a reference triangle and externally inscribe a square on the side . Now join the new vertices and of this square with the vertex , marking the points of intersection and . Next, draw lines perpendicular to the side through each of and . These points cross the sides and at and , respectively, resulting in an inscribed square
. The circumcircle through , ,
and
is then known as the Lucas -circles (Panakis 1973, p. 458; Yiu and Hatzipolakis 2001),
and repeating the process for other sides gives the corresponding - and -circles.

The Lucas -circle
has the beautiful trilinear center

where
is the area of the reference triangle is the circumradius
of the reference triangle , and radius

(Yiu and Hatzipolakis 2001).

The Lucas circles are pairwise tangent, although this fact seems to have been noted first only by Yiu and Hatzipolakis (2001).

There are two nonintersecting circles that are tangent to all three Lucas circles (these are the Soddy circles of the Lucas
central triangle ). The outer tangent circle is the circumcircle
of the reference triangle , while the inner
is the Lucas inner circle , which is the inverse
of the circumcircle in the Lucas
circles radical circle (P. Moses, pers. comm., Jan. 3, 2005).

There are also three circles analogous to the Lucas circles obtained when the original square is escribed instead of inscribed.

See also Lucas Central Triangle ,

Lucas Circles Radical Circle ,

Lucas
Tangents Triangle ,

Triangle Square Inscribing
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References Brisse, E. "6 Lucas Circles and Their Defining Square." http://pages.infinit.net/spqrsncf/fig02.htm . Moses,
P. "Circles and Triangle Centers Associated with the Lucas Circles." Forum
Geom. 5 , 97-106, 2005. http://forumgeom.fau.edu/FG2005volume5/FG200513index.html . Panakis,
I. Plane Trigonometry, Vol. 2. Published privately. Athens, Greece, 1973. Yiu,
P. Notes on Euclidean Geometry. 1998. http://www.math.fau.edu/yiu/EuclideanGeometryNotes.pdf . Yiu,
P. Introduction to the Geometry of the Triangle. 2002. http://www.math.fau.edu/yiu/GeometryNotes020402.ps . Yiu,
P. and Hatzipolakis, A. P. "The Lucas Circles of a Triangle."
Amer. Math. Monthly 108 , 444-446, 2001. http://www.math.fau.edu/yiu/monthly437-448.pdf . Referenced
on Wolfram|Alpha Lucas Circles
Cite this as:
Weisstein, Eric W. "Lucas Circles." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/LucasCircles.html

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