A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

LEMOINE'S, TUCKER'S, AND TAYLOR'S CIRCLES. 181
a2X a'
Hence DD'=- =
2A a2 +b2+ c2'
In like manner,
b3 C3
EE'= FF'=
a2 + b2 2^ 2 + b2 c2'
On account of this property, Mr. Tucker called the
Lemoine circle "' The Triplicate Ratio " circle.
CoR. 3. —The six triangles into which the Lemoine
hexagon is divided by lines from K to its angular points
are each similar to the triangle ABC.
COR. 4.-If lines drawn from the angles of a triangle
ABC, through a Brocardpoint, meet the circumcircle again
in A', B', C', the figure AB'CA'BC' is a Lemoine hexagon.
Prop. 2.-The radical axis of Lemoine's circle and the
circumeircle is the Pascal's line of the Lemoine hexagon.
Dem.-Let FE produced meet BC in X. Then
since FE' is antiparallel to BC, the points BFE'C are
concyclic. Hence the rectangle BX. CX = FX. E'X.
Therefore the radical axis of the Lemoine circle and
the circumcircle passes through X. Hence the proposition is proved.
COR. 1. —The polar of the symmedian point, with
respect to the Lemoine circle, is the Pascal's line of
the Lemoine hexagon.
For since DFE'D' is a quadrilateral inscribed in the
Lemoine circle, the polar of K passes through X. In
like manner, it passes through each pair of intersections of opposite sides.
COR. 2.-If the chords DE, D'E' intersect in p, EF,
E'F' in g, and FD, F'D' in r, the triangle pqr is in perspective with ABC.
Dem. —Join Ag, Cp, and let them meet in T; then
denoting the perpendiculars from T on the sides of
ABC by a, /3, y, respectively, we have a: f3:: perpendicular from p on BC: perpendicular from p on
o