## X(7) (GERGONNE 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           bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)
= sec2(A/2) : sec2(B/2) : sec2(C/2)

Barycentrics    1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)

Let A', B', C' denote the points in which the incircle meets the sides BC, CA, AB, respectively. The lines AA', BB', CC' concur in X(7).

X(7) is the {X(69),X(75)}-harmonic conjugate of X(8).

X(7) = reflection of X(I) in X(J) for these (I,J): (9,142), (144,9), (390,1), (673,1086), (1156,11)
X(7) = isogonal conjugate of X(55)
X(7) = isotomic conjugate of X(8)
X(7) = cyclocevian conjugate of X(7)
X(7) = inverse-in-incircle of (1323)
X(7) = complement of X(144)
X(7) = anticomplement of X(9)
X(7) = anticomplementary conjugate of X(329)
X(7) = X(I)-Ceva conjugate of X(J) for these (I,J): (75,347), (85,2), (86,77), (286,273), (331,278)

X(7) = cevapoint of X(I) and X(J) for these (I,J):
(1,57), (2,145), (4,196), (11,514), (55,218), (56,222), (63,224), (65,226), (81,229), (177,234)

X(7) = X(I)-cross conjugate of X(J) for these (I,J):
(1,2), (4,189), (11,514), (55,277), (56,278), (57,279), (65,57), (177,174), (222,348), (226,85), (354,1), (497,8), (517,88)

X(7) = crosspoint of X(I) and X(J) for these (I,J): (75,309), (86,286)
X(7) = crosssum of X(I) and X(J) for these (I,J): (41,1253), (42,228)
X(7) = crossdifference of any two points on line X(657)X(663)
X(7) = X(57)-Hirst inverse of X(1447)

X(7) = X(I)-beth conjugate of X(J) for these (I,J):
(7,269), (21,991), (75,75), (86,7), (99,7), (190,7), (290,7), (314,69), (645,344), (648,7), (662,6), (664,7), (666,7), (668,7), (670,7), (671,7), (811,264), (886,7), (889,7), (892,7), (903,7)

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