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Yff Points


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

\end{displaymath} (1)

This equation has a single real root $u$, which can by obtained by solving the Cubic Equation
\end{displaymath} (2)

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

The Isotomic Conjugate Point $U'$ is called the second Yff point. The Triangle Center Functions of the first and second points are given by
\alpha={1\over a}\left({c-u\over b-u}\right)^{1/3}
\end{displaymath} (6)

\alpha'={1\over a}\left({b-u\over c-u}\right)^{1/3},
\end{displaymath} (7)

respectively. Analogous to the inequality $\omega\leq\pi/6$ for the Brocard Angle $\omega$, $u\leq p/6$ holds for the Yff points, with equality in the case of an Equilateral Triangle. Analogous to
\end{displaymath} (8)

for $i=1$, 2, 3, the Yff points satisfy
\end{displaymath} (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

\overline{UU'}={4u\overline{IO}\Delta\over u^3+abc},
\end{displaymath} (10)

where $\Delta$ is the Area of the Triangle.

See also Brocard Points, Yff Triangles


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

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© 1996-9 Eric W. Weisstein