## X(74) (ISOGONAL CONJUGATE OF EULER INFINITY POINT)

 Interactive Applet

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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 A - 2 cos B cos C) : 1/(cos B - 2 cos C cos A) : 1/(cos C - 2 cos A cos B)
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[2a4 - (b2 - c2)2 - a2(b2 + c2)]

Barycentrics    a/(cos A - 2 cos B cos C) : b/(cos B - 2 cos C cos A) : c/(cos C - 2 cos A cos B)

As the isogonal conjugate of the point in which the Euler line meets the line at infinity, X(74) lies on the circumcircle.

X(74) = circumcircle-antipode of X(110)
X(74) is the point of intersection, other than A, B, and C of the circumcircle and Jerabek hyperbola.

In Hyacinthos 8129 (10/4/03), Floor van Lamoen noted that if X(74) is denoted by J, then each of the points A,B,C,J is J of the other three, in analogy with the well known property of orthocentric systems (that is, each of the points A,B,C,H is the orthocenter of the other three).

X(74) lies on these lines:
2,113    3,110    4,107    6,112    20,68    24,64    30,265    35,73    54,185    65,108    67,935    69,99    71,101    72,100    98,690    187,248    477,523    511,691    512,842    550,930

X(74) = reflection of X(I) in X(J) for these (I,J): (4,125), (110,3), (146,113), (399,1511)
X(74) = isogonal conjugate of X(30)
X(74) = complement of X(146)
X(74) = anticomplement of X(113)
X(74) = cevapoint of X(I) and X(J) for these (I,J): (15,16), (50,184)
X(74) = crosssum of X(I) and X(J) for these (I,J): (3,399), (616),617)
X(74) = X(I)-cross conjugate of X(J) for these (I,J): (186,54), (526,110)

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