HJB --- GMA --- UFF


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: (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 + 2 cos B cos C
Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

The Johnson Circle Theorem is the fact that if three congruent circles intersect in a point, then the circle passing through the other three intersections is congruent to them. This fourth circle is the Johnson circle of the three given circles. There are three congruent circles each tangent to two sides of triangle ABC. Peter Yff proved that their Johnson circle has center X(1478). The circle is here named the Johnson-Yff circle of the triangle.

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(1478) is the point C on page 5. See also X(495)-X(499).

X(1478) lies on these lines:
1,4    2,36    3,12    5,56    7,80    8,79    10,46    11,381    13,203    14,202    30,35    30,55    65,68    119,1470    148,192    442,958    474,1329    496,546    529,956    612,1370    908,997    975,1076    990,1074    1352,1469

X(1478) = reflection of X(I) in X(J) for these (I,J): (1,226), (55,495), (63,10)
X(1478) = isotomic conjugate of X(1121)
X(1478) = anticomplement of X(993)
X(1478) = X(1065)-Ceva conjugate of X(1)

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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