## 7221Variation on the Van Lamoen-Grinberg-Wolk-transform

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• Jun 1, 2003
Dear Hyacynthians,

looking for variations of the Van Lamoen-Grinberg-Wolk-transform I
discovered the following properties:

Consider a point P and a triangle ABC.
Let A'B'C' be the circumcevian triangle of P with respect to ABC.
Let A°, B° and C° be the circumcenters of the triangles PB'C', PC'A'
and PA'B'.

==> The lines AA°, BB° and CC° pass through a point S on the
circumcircle of ABC.

I found the following relationships using the ETC

P = Incenter X(1) ==> S = X(104)
P = Centroid X(2) ==> S = X(98)
P = Circumcenter X(3) ==> S = X(74)
P = Orthocenter X(4) ==> S = undetermined
P = Euler-center X(5) ==> S = X(1141)
P = Lemoine point X(6) ==> S = X(74)
P = Isogonal conjugate of Euler-center X(54) ==> S = X(74)

If we consider ABC as the circumcevian triangle of P with respect to
A'B'C' we have

Let A*, B* and C* be the circumcenters of the triangles PBC, PCA and
PAB.

==> The lines A'A*, B'B* and C'C* pass through a point T on the
circumcircle of ABC,

I found the following using the ETC

P = Incenter X(1) ==> T = undetermined
P = Centroid X(2) ==> T = X(1296)
P = Circumcenter X(3) ==> T = X(110)
P = Orthocenter X(4) ==> T = X(110)
P = Euler-center X(5) ==> T = X(1291)
P = Lemoine point X(6) ==> T = X(1296)
P = Isogonal conjugate of Euler-center X(54) ==> T = X(1291)

It seems that isogonal conjugates map to the same point.

Are these properties known ?

Greetings from Belgium

Eric Danneels
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