## X(2051) (EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE))

 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) = (s2 - r2)cos A - (s2 + r2)cos(B - C) - 2rs sin A (Peter J. C. Moses)
Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/[a3 - a(b2 - bc + c2) - bc(b + c)] (Paul Yiu)

X(2051) is the external center of similitude of the nine-point and Apollonius circles. Trilinears for the internal center, X(10), result from g(a,b,c) by changing the "-" just after "cos A" to "+". (The two circles are described just before X(1662).)

X(2051) lies on these lines:
2,573    4,386    5,10    6,2050    11,181    12,1682    27,275    43,1699    226,1465    321,908    469,2052    485,1685    486,1686    572,2185    1348,1693    1349,1694    1676,1683    1677,1684    1695,1698    1766,2339    2009,2019    2101,2020    2037,2040

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