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 _{}, select the center name from the list and, then, click on the vertices A, B and C successively.
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 1352,1469X(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)