## X(970) (CENTER OF THE APOLLONIUS CIRCLE)

 Interactive Applet

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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) = a[a3(b + c)2 + a(ab + ac - 2bc)(b2 + c2) - bc(b3 + c3) - a(b4 + c4) - (b5 + c5)]
= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (r2 - s2) cos A + 2rs sin A
Barycentrics    af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The Apollonius circle is described at X(182) as the circle tangent to the three excircles and encompassing them. That X(970) is its center was noted on New Year's Day, 2002, by Paul Yiu. (Hyacinthos #4619-4623)

X(970) lies on these lines: 1,181    3,6    5,10    21,51    40,43    185,411

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