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

X(1986)
(HATZIPOLAKIS REFLECTION 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           f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 + cos 2B + cos 2C) sin 3A csc 2A
Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let A'B'C' be the orthic triangle of triangle ABC. Let AB be the reflection of A in C', and define AC, BC, BA, CA, CB functionally. Then the nine-point circles of the triangles

AABAC,      BBCBA,      CCACB,     

concur in X(1986). (Antreas Hatzipolakis, Hyacinthos 7868, 9/12/03; coordinates by Barry Wolk, Hyacinthos 7876, 9/13/03)

X(1986) lies on these lines:
9,94    6,74    24,110    25,399    113,403    125,389    186,323    542,1843    648,1300    1844,1845

X(1986) = reflection of X(I) in X(J) for these (I,J): (4,1112), (74,974), (125,389)
X(1986) = X(4)-Ceva conjugate of X(403)
X(1986) = crosspoint of X(4) and X(186)
X(1986) = crosssum of X(3) and X(265)

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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