Tucker circles
The Tucker circles are a generalization of the cosine circle and first Lemoine circle.
Tucker circle
Let triangle be given.
is it’s circumcenter,
is it’s Lemoine point.
Let homothety centered at with factor
maps
into
.
Denote the crosspoints of sidelines these triangles as
Prove that points and
lies on the circle centered at
(Tucker circle).
Proof
is the parallelogram.
Denote
is antiparallel to
Similarly, is antiparallel to
is antiparallel to
is midpoint
is the midpoint
Similarly,
Let be the symmedian
through
It is known that three symmedians through are equal, so
is homothetic to
with center
and factor
So segments are tangents to
and points of contact are the midpoints of these segments.
Denote the circumcenter of
Therefore
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