Tucker Circles
Let three equal lines 
, 
, and 
be drawn Antiparallel to the sidesof a triangle so that two (say and ) are on the same side of the third line as 
. Then
is an isosceles Trapezoid, i.e., 
, 
, and 
are parallel to the respectivesides. The Midpoints 
, 
, and 
of the antiparallels are on the respective symmedians and divide themproportionally.
If 
divides 
in the same ratio, 
, 
, 
are parallel to the radii 
, 
,and 
and equal. Since the antiparallels are perpendicular to the symmedians, they are equal chords of a circlewith center which passes through the six given points. This circle is called the Tucker circle.
If
then the radius of the Tucker circle is
where
is the Brocard Angle.The Cosine Circle, Lemoine Circle, and Taylor Circle are Tucker circles.
See also Antiparallel, Brocard Angle, Cosine Circle, Lemoine Circle, Taylor Circle
References
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 271-277 and 300-301, 1929.
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