Yff's Conjecture

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Let $\triangle ABC$ be a triangle.

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


$8 \omega^3 < ABC$

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


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.