## X(1660) (1ST GRINBERG MIDPOINTS PERSPECTOR)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

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This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3[b8 + c8 - a8 - 2a2(b6 + c6)
+ 2b2c2(b4 + c4 - 5a4 + 3a2b2 + 3a2c2 - 3b2c2) + 2a6(b2 + c2)]

Barycentrics    af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let Ab be the point in which the line through A perpendicular to CA meets line BC, and define points Ac, Bc, Ba, Ca, Cb functionally. Let

Xa = midpoint{Ab, Ac},
Ya = midpoint{Ba, Ca},
Za = midpoint{Bc, Cb},

and define Xb, Xc, Yb, Yc, Zb, Zc functionally.

The lines AXa, BXb, CXc concur in X(20).
The lines AYa, BYb, CYc concur in X(393).
The lines AZa, BZb, CZc concur in X(6).
The lines XaYa, XbYb, XcYc concur in X(1660).
The lines YaZa, YbZb, YcZc concur in X(3).
The lines ZaXa, ZbXb, ZcXc concur in X(1661).

Contributed by Darij Grinberg, August 24, 2003; see Hyacinthos #7225.

X(1660) lies on these lines: 6,25    30,156    110,1370    394,1619    578,1596    1092,1498    1368,1503

X(1660) = midpoint of X(394) and X(1619)
X(1660) = X(20)-Ceva conjugate of X(577)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.