ъгли в триъгълник - тригонометрия

[Гост]

Ако α,β и γ са ъгли в триъгълник и 2tgα+tgβ+tgα*tgα*tg(α+γ) = 0, докажете,че α=γ.

Благодаря!

[svetoslav80]

[tex]tg\beta =tg(180-(\alpha +\gamma ))=-tg(\alpha +\gamma )[/tex]
[tex]tg(\alpha +\gamma ) = \frac{tg\alpha + tg\gamma}{1-tg\alpha tg\gamma }[/tex]

[tex]2tg\alpha+tg\beta+tg^2\alpha tg(\alpha+\gamma)=0[/tex]
[tex]2tg\alpha-tg(\alpha +\gamma )+tg^2\alpha tg(\alpha+\gamma)=0[/tex]
[tex]tg(\alpha+\gamma)(tg^2\alpha-1) + 2tg\alpha=0[/tex]
[tex]\frac{tg\alpha + tg\gamma}{1-tg\alpha tg\gamma }(tg^2\alpha-1) + 2tg\alpha=0[/tex]
[tex]\frac{(tg\alpha+tg\gamma)(tg^2\alpha-1)+2tg\alpha(1-tg\alpha tg\gamma)}{1-tg\alpha tg\gamma }=0[/tex]
[tex](tg\alpha+tg\gamma)(tg^2\alpha-1)+2tg\alpha(1-tg\alpha tg\gamma)=0[/tex]
[tex]tg^3\alpha-tg\alpha+tg\gamma tg^2\alpha-tg\gamma+2tg\alpha-2tg^2\alpha tg\gamma=0[/tex]
[tex]tg^3\alpha+tg\alpha-tg\gamma tg^2\alpha-tg\gamma=0[/tex]
[tex]tg^2\alpha(tg\alpha-tg\gamma)+tg\alpha-tg\gamma=0[/tex]
[tex](tg\alpha-tg\gamma)(tg^2\alpha+1)=0[/tex]
[tex]tg\alpha-tg\gamma=0[/tex]
[tex]tg\alpha=tg\gamma[/tex]
[tex]\alpha=\gamma[/tex]