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Lucas Inner Circle


LucasCirclesTangentCircles

There are two nonintersecting circles that are tangent to all three Lucas circles. (These are therefore the Soddy circles of the Lucas central triangle.) The inner one, dubbed the Lucas inner circle here for the first time, is the inverse of the circumcircle in the Lucas circles radical circle (P. Moses, pers. comm., Jan. 3, 2005).

Its center has triangle center function

 alpha=a(bccosA+8Delta),
(1)

where Delta is the area of the reference triangle, which lies on the Brocard axis, and its radius is

R_I=(abc)/(4[(a^2+b^2+c^2)+7Delta])
(2)
=R/(4cotomega+7),
(3)

where omega is the Brocard angle (P. Moses, pers. comm., Jan. 3, 2005).

It has circle function

 l=-(2bc)/((a^2+b^2+c^2)+7Delta),
(4)

corresponding to the triangle centroid G, which is Kimberling center X_2.

LucasInnerCircleOrthogonal

It is orthogonal to the Parry circle.

The circumcircle, Lucas circles radical circle, Lucas inner circle, and Brocard circle share the Lemoine axis as their radical line and hence are part of the Schoute coaxal system (P. Moses, pers. comm., Jan. 3, 2005).


See also

Lucas Circles, Lucas Circles Radical Circle, Lucas Inner Triangle

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Cite this as:

Weisstein, Eric W. "Lucas Inner Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LucasInnerCircle.html

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