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Johnson-Yff Circles


Yff-JohnsonCircles

Since each triplet of Yff circles are congruent and pass through a single point, they obey Johnson's theorem. As a result, in each case, there is a fourth circle congruent to the original three and passing through the points of pairwise intersection. These circles have radii

rho_1=(rR)/(R+r)
(1)
rho_2=(rR)/(R-r),
(2)

and their centers are

alpha_(1478)=1+2cosBcosC
(3)
alpha_(1479)=1-2cosBcosC,
(4)

which are Kimberling centers X_(1478) and X_(1479), respectively.

The circle functions of the Johnson circles do not correspond to any Kimberling centers, and the Johnson-Yff circles do not pass through any Kimberling centers.

The sets of points (Y_A, Y_B, Y_C, X_(1478)) and (Z_A, Z_B, Z_C, X_(1479)) comprise two orthocentric systems.


See also

Johnson Circles, Johnson's Theorem, Orthocentric System, Yff Circles

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References

Kimberling, C. "Encyclopedia of Triangle Centers: X(1478)=Center of Johnson-Yff Circle." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X1478.

Referenced on Wolfram|Alpha

Johnson-Yff Circles

Cite this as:

Weisstein, Eric W. "Johnson-Yff Circles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Johnson-YffCircles.html

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