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

X(495)
(JOHNSON MIDPOINT)


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           2 + cos(B - C) : 2 + cos(C - A) : 2 + cos(A - B)
Barycentrics    (sin A)[2 + cos(B - C)] : (sin B)[2 + cos(C - A)] : (sin C)[2 + cos(A - B)]

X(495) is the midpoint of segments C1-to-P1, C2-to-P2, C3-to-P3 in the Johnson four-circle configuration.

Roger A. Johnson, Advanced Euclidean Geometry, Dover, New York, 1960, page 75.

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(495) is the point R on page 5. (See also X(496)-X(499) and X(1478), X(1479).)

X(495) lies on these lines:
1,5    2,956    3,388    4,390    8,442    10,141    30,55    35,550    36,549    56,140    202,395    203,396    226,517    381,497    392,908    429,1068    529,993    612,1060

X(495) = complement of X(956)

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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