# Yff's Conjecture

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## Theorem

Let $\triangle ABC$ be a triangle.

Let $\omega$ be the Brocard angle of $\triangle ABC$.

Then:

- $8 \omega^3 < ABC$

where $A, B, C$ are measured in radians.

## Proof

The Abi-Khuzam Inequality states that

- $\sin A \cdot \sin B \cdot \sin C \le \paren {\dfrac {3 \sqrt 3} {2 \pi} }^3 A \cdot B \cdot C$

The maximum value of $A B C - 8 \omega^3$ occurs when two of the angles are equal.

So taking $A = B$, and using $A + B + C = \pi$, the maximum occurs at the maximum of:

- $\map f A = A^2 \paren {\pi - 2 A} - 8 \paren {\map \arccot {2 \cot A - \cot 2 A} }^3$

which occurs when:

- $2 A \paren {\pi - 3 A} - \dfrac {48 \paren {\map \arccot {\frac 1 2 \paren {3 \cot A + \tan A} } }^2 \paren {1 + 2 \cos 2 A} } {5 + 4 \cos 2 A} = 0$

## Also known as

Can also be seen referred to as the **Yff conjecture**.

## Source of Name

This entry was named for Peter Yff.

## Historical Note

Peter Yff made this conjecture in a paper of $1963$.

It was proved by Faruk Fuad Abi-Khuzam in $1974$, and reiterated by him and Artin B. Boghossian in $1989$.

Hence despite it no longer being a conjecture, it is still referred to as one.

## Sources

- 1963: Peter Yff:
*An Analog of the Brocard Points*(*Amer. Math. Monthly***Vol. 70**: 495 – 501) www.jstor.org/stable/2312058 - 1974: Faruk F. Abi-Khuzam:
*Proof of Yff's Conjecture on the Brocard Angle of a Triangle*(*Elem. Math.***Vol. 29**: 141 – 142) - 1983: François Le Lionnais and Jean Brette:
*Les Nombres Remarquables*... (previous) ... (next): $0,56559 56245 \ldots$ - 1989: Faruk F. Abi-Khuzam and Artin B. Boghossian:
*Some Recent Geometric Inequalities*(*Amer. Math. Monthly***Vol. 96**: 576 – 589) www.jstor.org/stable/2325176

- Weisstein, Eric W. "Yff Conjecture." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/YffConjecture.html