## X(112) (Ψ(ORTHOCENTER, SYMMEDIAN 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.

 The JRE (Java Runtime Environment) is not enabled in your browser!

This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           a/(sin 2B - sin 2C) : b/(sin 2C - sin 2A) : c/(sin 2A - sin 2B)
= f(a,b,c) : f(b,c,a) : f(c,a,b) where f(a,b,c) = a/[(b2 - c2)(b2 + c2 - a2)]

Barycentrics    a2/(sin 2B - sin 2C) : b2/(sin 2C - sin 2A) : c2/(sin 2A - sin 2B)

X(112) = Ψ(X(4), X(6))

X(112) lies on these lines:
2,127    4,32    6,74    19,759    25,111    27,675    28,105    33,609    50,477    54,217    58,103    99,648    100,162    102,284    104,1108    109,163    186,187    230,403    250,691    251,427    286,767    376,577    393,571    523,935    789,811

X(112) = reflection of X(I) in X(J) for these (I,J): (4,132), (1297,3)
X(112) = isogonal conjugate of X(525)
X(112) = anticomplement of X(127)
X(112) = X(I)-Ceva conjugate of X(J) for these (I,J): (249,24), (250,25)
X(112) = cevapoint of X(I) and X(J) for these (I,J): (32,512), (427,523)
X(112) = X(I)-cross conjugate of X(J) for these (I,J): (25,250), (512,4), (523,251)
X(112) = crossdifference of any two points on line X(122)X(125)
X(112) = barycentric product of X(1113) and X(1114)

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