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Tucker Circles

Let three equal lines $P_1Q_1$, $P_2Q_2$, and $P_3Q_3$ be drawn Antiparallel to the sides of a triangle so that two (say $P_2Q_2$ and $P_3Q_3$) are on the same side of the third line as $A_2P_2Q_3A_3$. Then $P_2Q_3P_3Q_2$ is an isosceles Trapezoid, i.e., $P_3Q_2$, $P_1Q_3$, and $P_2Q_1$ are parallel to the respective sides. The Midpoints $C_1$, $C_2$, and $C_3$ of the antiparallels are on the respective symmedians and divide them proportionally.

If $T$ divides $KO$ in the same ratio, $TC_1$, $TC_2$, $TC_3$ are parallel to the radii $OA_1$, $OA_2$, and $OA_3$ and equal. Since the antiparallels are perpendicular to the symmedians, they are equal chords of a circle with center $T$ which passes through the six given points. This circle is called the Tucker circle.


c\equiv {\overline{KC_1}\over\overline{KA_1}}={\overline{KC_...

then the radius of the Tucker circle is


where $\omega$ 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


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.

© 1996-9 Eric W. Weisstein