## Tucker Circles

Let three equal lines , , and be drawn Antiparallel to the sides of 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 respective sides. The Midpoints , , and of the antiparallels are on the respective symmedians and divide them proportionally.

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 circle with 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.