Chủ đề này có 1 trả lời

[DOTOANNANG]

Chứng minh rằng $\it{,}$ với $\it{\Delta }\,\text{ABC}$ thì $\it{:}$

$\sum\,\it{\sin\,A}\,\,\,\it{\sin}\,\it{(}\,\,\it{A}- \it{B}\,\,\it{)}\,\,\,\it{\sin}\,\it{(}\,\,\it{A}- \it{C}\,\,\it{)}\geqq \it{0}$

 

[DOTOANNANG]

Chứng minh rằng $\it{,}$ với $\it{\Delta }\,\text{ABC}$ thì $\it{:}$

$\sum\,\it{\sin\,A}\,\,\,\it{\sin}\,\it{(}\,\,\it{A}- \it{B}\,\,\it{)}\,\,\,\it{\sin}\,\it{(}\,\,\it{A}- \it{C}\,\,\it{)}\geqq \it{0}$

$\Leftrightarrow \,\,\frac{\it{(}\,\,\sin ^{\,\it{2}}\it{B}- \sin ^{\,\it{2}}\it{A}\,\,\it{)}\,\it{(}\,\,\sin ^{\,\it{2}}\it{A}- \sin ^{\,\it{2}}\it{C}\,\,\it{)}}{\sin ^{\,\it{2}}\it{B}\,\sin ^{\,\it{2}}\it{C}}+$ $\frac{\it{(}\,\,\sin ^{\,\it{2}}\it{C}- \sin ^{\,\it{2}}\it{B}\,\,\it{)}\,\it{(}\,\,\sin ^{\,\it{2}}\it{B}- \sin ^{\,\it{2}}\it{A}\,\,\it{)}}{\sin ^{\,\it{2}}\it{C}\,\sin ^{\,\it{2}}\it{A}}+$ $\frac{\it{(}\,\,\sin ^{\,\it{2}}\it{A}- \sin ^{\,\it{2}}\it{C}\,\,\it{)}\,\it{(}\,\,\sin ^{\,\it{2}}\it{C}- \sin ^{\,\it{2}}\it{B}\,\,\it{)}}{\sin ^{\,\it{2}}\it{A}\,\sin ^{\,\it{2}}\it{B}}$ $\leqq \it{0}$

$\lceil$ https://diendantoanh...-0/#entry704842 $\rfloor$

Spoiler

$\it{[}$ Schur decomposition type $\it{!}$   $\it{]}$    $\sum\,\it{\sin\,A}\it{\sin}\,\it{(}\,\,\it{A}- \it{B}\,\,\it{)}\it{\sin}\,\it{(}\,\,\it{A}- \it{C}\,\,\it{)}\leqq \it{(}\,\,\it{A}- \it{B}\,\,\it{)}\it{(}\,\,\it{A}- \it{C}\,\,\it{)}\it{\sin\,A}$