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

X(115)
(CENTER OF KIEPERT HYPERBOLA)


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(b2 - c2)2 : ca(c2 - a2)2 : ab(a2 - b2)2
                                    = cos A - 2 cos(B - C) + cot ω sin A (Peter J. C. Moses, 9/12/03)

Barycentrics    (b2 - c2)2 : (c2 - a2)2 : (a2 - b2)2

X(115) lies on the nine-point circle
X(115) = X(99)-of-medial triangle
X(115) = X(101)-of-orthic triangle

Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.

X(115) lies on these lines:
2,99    4,32    5,39    6,13    11,1015    30,187    50,231    53,133    76,626    116,1086    120,442    125,245    127,338    128,233    129,389    131,216    232,403    316,385    325,538    395,530    396,531    593,1029    804,1084

X(115) = midpoint of X(I) and X(J) for these (I,J): (4,98), (13,14), (99,148), (316,385)
X(115) = reflection of X(I) in X(J) for these (I,J): (99,620), (114,5), (187,230), (325,625)
X(115) = isogonal conjugate of X(249)
X(115) = inverse-in-orthocentroidal-circle of X(6)
X(115) = complementary conjugate of X(512)
X(115) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,523), (4,512), (338,125)
X(115) = crosspoint of X(I) and X(J) for these (I,J): (2,523), (68,525)
X(115) = crosssum of X(I) and X(J) for these (I,J): (6,110), (24,112), (163,849)
X(115) = crossdifference of any two points on line X(110)X(351)
X(115) = X(2)-Hirst inverse of X(148)

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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