INSTITUTO DE MATEMÁTICA
HJB --- GMA --- UFF

X(7)
(GERGONNE POINT)


Click here to access the list of all triangle centers.

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 Run Macro Tool, select the center name from the list and, then, click on the vertices A, B and C successively.

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Download all construction files and macros: tc.zip (10.1 Mb).
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) lies on these lines:
1,20    2,9    3,943    4,273    6,294    8,65    11,658    12,1268    21,56    27,81    37,241    33,1041    34,1039    58,272    59,1275    72,443    73,1246    76,1479    80,150    92,189    100,1004    104,934    108,1013    109,675    145,1266    171,983    174,234    177,555    190,344    192,335    193,239    218,277    220,1223    225,969    238,1471    253,280    256,982    274,959    281,653    286,331    310,314    330,1432    349,1269    354,479    404,1259    452,1467    464,1214    480,1376    492,1267    513,885    517,1000    528,664    554,1082    594,599    604,1429    757,1414    840,927    857,1901    870,1431    940,1407    941,1427    944,1389    952,1159    986,1254    987,1106    1002,1362    1020,1765    1061,1870    1354,1367    1365,1366    1386,1456    1419,1449    1435,1848    1486,1602    1617,1621

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.

If you have questions or suggestions, please, contact us using the e-mail presented here.

Departamento de Matemática Aplicada -- Instituto de Matemática -- Universidade Federal Fluminense



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