## X(2509) (RADICAL CENTER OF {NINE-POINT CIRCLE, 1ST LEMOINE 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) = (b - c)UV,
where U = a3 - a2(b + c) + a(b + c)2 - (b + c)(b2 + c2),
where V = a3 + a2(b + c) + a(b + c)2 + (b + c)(b2 + c2)

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

X(2509) lies on these lines: 6,521    44,513    525,2485    905,918    926,2494

X(2509) = midpoint of X(661) and X(2484)
X(2509) = crosspoint of X(2) and X(1783)
X(2509) = crosssum of X(I) and X(J) for these I,J: 1,2509    6,905

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