cho tam giác ABC có
BC = a
AC = b
AB = c
biết \(\dfrac{ab}{b+c}+\dfrac{bc}{c+d}+\dfrac{ca}{a+b}=\dfrac{ca}{b+c}+\dfrac{ab}{c+a}+\dfrac{bc}{a+b}\)
cmr tam giác ABC cân
Lời giải:
\(\frac{ab}{b+c}+\frac{bc}{c+a}+\frac{ca}{a+b}=\frac{ca}{b+c}+\frac{ab}{c+a}+\frac{bc}{a+b}\)
\(\Leftrightarrow \frac{a(b-c)}{b+c}+\frac{b(c-a)}{c+a}+\frac{c(a-b)}{a+b}=0\)
\(\Leftrightarrow \frac{a(b-c)}{b+c}-\frac{b(b-c+a-b)}{c+a}+\frac{c(a-b)}{a+b}=0\)
\(\Leftrightarrow \frac{a(b-c)}{b+c}-\frac{b(b-c)}{c+a}-\frac{b(a-b)}{c+a}+\frac{c(a-b)}{a+b}=0\)
\(\Leftrightarrow (b-c)\left(\frac{a}{b+c}-\frac{b}{c+a}\right)-(a-b)\left(\frac{b}{c+a}-\frac{c}{a+b}\right)=0\)
\(\Leftrightarrow (b-c).\frac{(a-b)(a+b+c)}{(b+c)(c+a)}-(a-b).\frac{(b-c)(b+c+a)}{(c+a)(a+b)}=0\)
\(\Leftrightarrow (a+b+c)(a-b)(b-c)\left(\frac{1}{(b+c)(c+a)}-\frac{1}{(c+a)(a+b)}\right)=0\)
\(\Leftrightarrow (a+b+c)(a-b)(b-c).\frac{a-c}{(a+b)(b+c)(c+a)}=0\)
Vì $a,b,c$ là 3 cạnh tam giác nên \(\frac{a+b+c}{(a+b)(b+c)(c+a)}\neq 0\)
Do đó: \((a-b)(b-c)(a-c)=0\Rightarrow \left[\begin{matrix} a=b\\ b=c\\ c=a\end{matrix}\right.\)
Suy ra tam giác $ABC$ cân
Ta có đpcm.
