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Yff Center of Congruence

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YffCenter

Begin with the Yff central triangle and then parallel-displace the isoscelizers in such a way as to reduce the central triangle to a point while keeping the other three triangles congruent to each other. The point thus constructed is called the Yff center of congruence, and has triangle center function

alpha=sec(1/2A).
(1)
YffCenterCons

By analogy with the determination of the Yff central triangle, the angle alpha_1 is related to the isoscelizer distance l_1 and the inner triangle side lengths t_i are given by

sin(1/2alpha_1)=sqrt((1-cosalpha_1)/2)=(1/2(t_2+t_3))/(l_1)
(2)

and so on. Therefore, the length l_i and t_i can be determined by solving the six simultaneous equations

l_2+l_3-t_1=s_1
(3)
l_1+l_3-t_2=s_2
(4)
l_1+l_2-t_3=s_3
(5)
((t_2+t_3)/(l_1))^2=2(1-(s_2^2+s_3^2-s_1^2)/(2s_2s_3))
(6)
((t_1+t_3)/(l_2))^2=2(1-(s_1^2+s_3^2-s_2^2)/(2s_1s_3))
(7)
((t_1+t_2)/(l_3))^2=2(1-(s_1^2+s_2^2-s_3^2)/(2s_1s_2)).
(8)

See also

Congruent Isoscelizers Point, Isoscelizer, Yff Central Triangle

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References

Kimberling, C. "Yff Center of Congruence." http://faculty.evansville.edu/ck6/tcenters/recent/yffcc.html.

Referenced on Wolfram|Alpha

Yff Center of Congruence

Cite this as:

Weisstein, Eric W. "Yff Center of Congruence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/YffCenterofCongruence.html

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