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 line 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 extended sides in six points, then those six points are
concyclic. The center of
the resulting circle lies on the Brocard
axis and divides the directed line segment from
to 
in the ratio 
. This circle is not a Tucker circle.
The first Lemoine circle also occurs in the 36 cataloged six-piece triangle dissections of a generic non-right triangle into
similar triangles. Twelve layouts, nine with
junctions at vertices of the first
Brocard triangle and three at the symmedian point,
form four cyclic groups with the same three cuts through 
parallel to the sides.
Their six endpoints lie on the first Lemoine circle. Six of these individual layouts
have all four boundary cut points on this circle.
The other three concyclic six-piece layouts have junctions
at vertices of the third
Brocard triangle and give three different circles,
whose centers also lie on the Brocard
axis (E. Pegg, Jr., pers. comm., Oct. 3, 2026).
The first Lemoine circle is a special case of a Tucker circle.
