## X(481) (1ST EPPSTEIN POINT)

 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) = 1 - 2 sec A/2 cos B/2 cos C/2
= 1 - 4(area)/[a(b + c - a)] : 1 - 4(area)/[b(c + a - b)] : 1 - 4(area)/[c(a + b - c)] [E. Brisse, 3/20/01]
Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let S be the inner Soddy circle and Sa, Sb, Sc the Soddy circles tangent to S. Let Ia = S∩Sa, Ea = Sb∩Sc, and determine Ib, Ic, Eb, Ec cyclically. Then X(481) is the point of concurrence of lines Ia-to-Ea, Ib-to-Eb, Ic-to-Ec.

David Eppstein, "Tangent spheres and triangle centers," American Mathematical Monthly, 108 (2001) 63-66.

X(481) lies on these lines: 1,7    174,1127    226,485

X(481) = X(79)-Ceva conjugate of X(482)

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