X(2021) (RADICAL TRACE OF GALLATLY CIRCLE AND CIRCUMCIRCLE)

 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) = 3 sin(A - ω) - 2 sin(A + ω) + sin(A + 3ω)
Trilinears           f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a(b6 + c6 + 3a4b2 + 3a4c2 - 2a2b4 - 2a2c4 - 2a2b2c2 - b2c4 - b4c2)       (M. Iliev, 5/13/07)

Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2021) lies on these lines:
3,6    30,2025    115,1513    538,1569    620,736    625,1506

X(2021) = midpoint of X(39) and X(187)
X(2021) = reflection of X(2025) in X(2024)

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