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Yff Points


YffPoints

Let points A^', B^', and C^' be marked off some fixed distance x along each of the sides BC, CA, and AB. Then the lines AA^', BB^', and CC^' concur in a point U known as the first Yff point if

 x^3=(a-x)(b-x)(c-x).
(1)

This equation has a single real root u, which can by obtained by solving the cubic equation

 f(x)=2x^3-px^2+qx-r=0,
(2)

where

p=a+b+c
(3)
q=ab+ac+bc
(4)
r=abc.
(5)

The isotomic conjugate U^' is called the second Yff point. The triangle center functions of the first and second points are given by

 alpha=1/a((c-u)/(b-u))^(1/3)
(6)

and

 alpha^'=1/a((b-u)/(c-u))^(1/3),
(7)

respectively.

Analogous to the inequality omega<=pi/6 for the Brocard angle omega, u<=p/6 holds for the Yff points, with equality in the case of an equilateral triangle. Analogous to

 omega<alpha_i<pi-3omega
(8)

for i=1, 2, 3, the Yff points satisfy

 u<a_i<p-3u.
(9)

Yff (1963) gives a number of other interesting properties. The line UU^' is perpendicular to the line containing the incenter I and circumcenter O, and its length is given by

 UU^'^_=(4uIO^_Delta)/(u^3+abc),
(10)

where Delta is the area of the triangle.

The Cevian triangles of the Yff points are known as the Yff triangles.


See also

Brocard Points, First Yff Triangle, Second Yff Triangle

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References

Yff, P. "An Analog of the Brocard Points." Amer. Math. Monthly 70, 495-501, 1963.

Referenced on Wolfram|Alpha

Yff Points

Cite this as:

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

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