First Lemoine Circle

The first Lemoine circle is obtained by drawing lines , , and through the symmedian point and parallel to the sides of the triangle . The points where the parallel lines intersect the sides of lie on this circle, which is sometimes called the triplicate-ratio circle (Tucker 1883; Kimberling 1998, p. 233).

This circle has circle function

(1)

corresponding to Kimberling center , which is the complement of the symmedian point. It has center at the Brocard midpoint , i.e., the midpoint of , where is the circumcenter and is the symmedian point, and radius

			(2)
			(3)

where is the circumradius, is the inradius, and is the Brocard angle of the original triangle (Johnson 1929, p. 274).

Kimberling centers and (the intersections with the Brocard axis) lie on the first Lemoine circle.

The first Lemoine circle and Brocard circle are concentric, and the triangles , , and are similar to (Tucker 1883).

The first Lemoine circle divides any side into segments proportional to the squares of the sides

(4)

Furthermore, the chords cut from the sides by the Lemoine circle are proportional to the squares of the sides.

Płatek (2026) gave another Lemoine-type circle construction. If , , and are the intersections of , , and with the circumcircle of , and the three circles through , , and tangent to the circumcircle at , , and , respectively, meet the opposite sidelines in six points, then those six points are concyclic. The center of the resulting circle lies on the Brocard axis and divides the directed segment from to in the ratio . This circle is not a Tucker circle.

The first Lemoine circle is a special case of a Tucker circle.