## Euler-Vecten-Gibert Points, X(2043) and X(2044)

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• Dear Hyacinthists, Let ABC be a triangle, X a point and k a real number. The k-dilation of ABC from X is denoted by A B C . The k-dilation of B and C from A
Message 1 of 1 , Jan 6, 2013
Dear Hyacinthists,

Let ABC be a triangle, X a point and k a real number.
The k-dilation of ABC from X is denoted by A'B'C'.
The k-dilation of B' and C' from A is denoted by Ba and Ca;
the k-dilation of C' and A' from B is denoted by Cb and Ab;
and the k-dilation of A' and B' from C is denoted by Ac and Bc.

Let C(X,t) be the ellipse which pass through Ba, Ca, Cb, Ab, Ac and Bc.

Denoted the internal center of similitude of C(X,t) and the Steiner circumellipse by Pi(X,t); and external center of similitude by Pe(X,t)

If X(x:y:z) (barycentric coordinates),

Pi(X,k), (1+k^2+Sqrt[1-2k+k^3+k^4])x + k(y+z): ... : ...

Pe(X,k), (1+k^2-Sqrt[1-2k+k^3+k^4])x + k(y+z): ... : ...

In particular, if X=X(3) and k=-1:

Pi(X(3),-1)= X(2043), (2+Sqrt[3])a^2SA - b^2SB - c^2SC:...:...

Pe(X(3),-1)= X(2044) (2-Sqrt[3])a^2SA - b^2SB - c^2SC:...:...

Is there a similar geometric interpretation for the other points:
Euler-Vecten-Gibert Points, X(2041) - X(2046)?

Best regards,
Angel Montesdeoca
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