## X(23) (FAR-OUT 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) = a[b4 + c4 - a4 - b2c2]
Barycentrics    af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics    2 sin 2A - 3 tan ω : 2 sin 2B - 3 tan ω : 2 sin 2C - 3 tan ω       (M. Iliev, 5/13/07)

X(23) is the inverse-in-circumcircle of X(2).

X(23) lies on these lines:
2,3    6,353    51,575    52,1614    94,98    105,1290    110,323    111,187    143,1199    159,193    184,576    232,250    251,1194    324,1629    385,523    477,1302    895,1177    1196,1627    1297,1804

X(23) is the {X(22),X(25)}-harmonic conjugate of X(2).

X(23) = reflection of X(I) in X(J) for these (I,J): (110,1495), (323,110), (691,187), (858,468)
X(23) = isogonal conjugate of X(67)
X(23) = inverse-in-circumcircle of X(2)
X(23) = anticomplement of X(858)
X(23) = crosspoint of X(111) and X(251)
X(23) = crosssum of X(I) and X(J) for these (I,J): (125,690), (141,524)
X(23) = crossdifference of any two points on line X(39)X(647)

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