## X(12) ({X(1),X(5)}-HARMONIC CONJUGATE OF X(11))

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
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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           1 + cos(B - C) : 1 + cos(C - A) : 1 + cos(A - B)
= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos2(B/2 - C/2)
= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c)2/(b + c - a)

Barycentrics    a(1 + cos(B - C)) : b(1 + cos(C - A)) : c(1 + cos(A - B))
= h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = (b + c)2/(b + c - a)

Let A'B'C' be the touch points of the nine-point circle with the A-, B-, C- excircles, respectively. The lines AA', BB', CC' concur in X(12).

X(12) is the {X(1),X(5)}-harmonic conjugate of X(11).

X(12) = isogonal conjugate of X(60)
X(12) = isotomic conjugate of X(261)
X(12) = complement of X(2975)
X(12) = X(10)-Ceva conjugate of X(201)
X(12) = crosssum of X(58) and X(1437)
X(12) = X(I)-beth conjugate of X(J) for these (I,J): (10,12), (1089,1089)

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