Yff's Conjecture

From ProofWiki
Jump to navigation Jump to search

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:

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


This needs considerable tedious hard slog to complete it.
In particular: Needs expanding and completing
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Finish}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


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

Retrieved from "https://proofwiki.org/w/index.php?title=Yff%27s_Conjecture&oldid=433107"