Lucas Circles


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

The Lucas A-circle has the beautiful trilinear center


where Delta is the area of the reference triangle R 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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Brisse, E. "6 Lucas Circles and Their Defining Square.", P. "Circles and Triangle Centers Associated with the Lucas Circles." Forum Geom. 5, 97-106, 2005., I. Plane Trigonometry, Vol. 2. Published privately. Athens, Greece, 1973.Yiu, P. Notes on Euclidean Geometry. 1998., P. Introduction to the Geometry of the Triangle. 2002., P. and Hatzipolakis, A. P. "The Lucas Circles of a Triangle." Amer. Math. Monthly 108, 444-446, 2001.

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Lucas Circles

Cite this as:

Weisstein, Eric W. "Lucas Circles." From MathWorld--A Wolfram Web Resource.

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