Question

cho tam giác ABC có BC=a, AB=c, AC=b. Tam giác ABC có đặc điểm gì nếu

\(\frac{1+CosB}{sinB}=\frac{2a+c}{\sqrt{4a^2-c^2}}\)

Answer 1

\(\frac{1+cosB}{\sqrt{1-cos^2B}}=\frac{2a+c}{\sqrt{4a^2-c}}\Leftrightarrow\sqrt{\frac{1+cosB}{1-cosB}}=\frac{2a+c}{\sqrt{4a^2-c^2}}\)

\(\Leftrightarrow\frac{1+cosB}{1-cosB}=\frac{4a^2+4ac+c^2}{4a^2-c^2}\)

\(\Leftrightarrow4a^2-c^2+\left(4a^2-c^2\right)cosB=4a^2+4ac+c^2-\left(4a^2+4ac+c^2\right)cosB\)

\(\Leftrightarrow\left(4a^2+2ac\right)cosB=c^2+2ac\)

\(\Leftrightarrow cosB=\frac{c^2+2ac}{4a^2+2ac}=\frac{c\left(c+2a\right)}{2a\left(c+2a\right)}=\frac{c}{2a}\)

\(\Leftrightarrow\frac{a^2+c^2-b^2}{2ac}=\frac{c}{2a}\Leftrightarrow a^2+c^2-b^2=c^2\)

\(\Leftrightarrow a=b\Rightarrow\) tam giác cân tại C