Lời giải:
Nếu $a,b,c$ lập thành csc thì $b=a+m, c=a+2m$ với $m$ là công sai.

Khi đó:

$3(a^2+b^2+c^2)-6(a-b)^2=3[a^2+(a+m)^2+(a+2m)^2]-6(a-a-m)^2$

$=3(a^2+a^2+m^2+2am+a^2+4m^2+4am)-6m^2$

$=3(3a^2+5m^2+6am)=9a^2+15m^2+18am-6m^2$

$=9a^2+9m^2+18am$

$=9(a^2+m^2+2am)=9(a+m)^2=(3a+3m)^2$

$=(a+a+m+a+2m)^2=(a+b+c)^2$ (đpcm).

Theo định lí cos ta có: \({a^2} = {b^2} + {c^2} - 2bc\;\cos A\)

\( \Rightarrow {b^2} + {c^2} - {a^2} = 2bc\;\cos A\)(1)

a) Nếu góc A nhọn thì \(\cos A > 0\)

Từ (1), suy ra \({b^2} + {c^2} > {a^2}\)

b) Nếu góc A tù thì \(\cos A < 0\)

Từ (1), suy ra \({b^2} + {c^2} < {a^2}\)

c) Nếu góc A vuông thì \(\cos A = 0\)

Từ (1), suy ra \({b^2} + {c^2} = {a^2}\)

Từ \(1=a+b+c\Rightarrow1=\left(a+b+c\right)^2\le3\left(a^2+b^2+c^2\right).\)(bất đẳng thức bunhiacopxki)

\(\Leftrightarrow a^2+b^2+c^2\ge\frac{1}{3}\)(*)

Ta có  : \(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2\le3\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\)(1)

Dễ thấy \(\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=3+\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{b}{b}+\frac{c}{a}+\frac{a}{c}\)

\(\ge3+2\sqrt{\frac{a}{b}.\frac{b}{a}}+2\sqrt{\frac{c}{b}\frac{b}{c}}+2\sqrt{\frac{a}{c}\frac{c}{a}}=3+2+2+2=9\)(bất đẳng thức cô si)

\(Hay:\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge9\left(do:a+b+c=1\right)\)(2)

Từ (1) và (2) suy ra \(9^2\le\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2\le3\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\)

\(\Rightarrow\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\ge27\)(**)

Ta có \(\left(a+\frac{1}{a}\right)^2+\left(b+\frac{1}{b}\right)^2+\left(c+\frac{1}{c}\right)^2\)

\(=a^2+2+\frac{1}{a^2}+b^2+2+\frac{1}{b^2}+c^2+2+\frac{1}{c^2}\)

\(=\left(a^2+b^2+c^2\right)+\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)+6\)

\(\ge\frac{1}{3}+27+6=33+\frac{1}{3}>33\)(theo (*) và (**) )

\(a^2+b^2+c^2=ab+bc+ca\)

\(\Rightarrow2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)

\(\Rightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ca+a^2\right)=0\)

\(\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)

\(\Rightarrow\hept{\begin{cases}\left(a-b\right)^2=0\\\left(b-c\right)^2=0\\\left(c-a\right)^2=0\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}a-b=0\\b-c=0\\c-a=0\end{cases}\Leftrightarrow a=b=c}\)

TL:

1) 

Ta có:  \(2a^2+2b^2+2c^2=2ab+2ac+2bc\) 

          \(2a^2+2b^2+2c^2-2ab-2ac-2bc=0\)  

        \(\left(a^2-2ab+b^2\right)+\left(a^2-2ac+c^2\right)+\left(b^2-2bc+c^2\right)=0\) 

        \(\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\) 

\(\Rightarrow\left(a-b\right)^2=0\) và\(\left(a-c\right)^2=0\)  và  \(\left(b-c\right)^2=0\) 

\(\Rightarrow a-b=0\) và \(â-c=0\) và  \(b-c=0\) 

=>a=b=c(đpcm)

hình như câu B đề sai bạn nhé!

mk sửa lại ko biết có đúng ko:)

\(â^2+b^2+c^2=2\left(a+b+c\right)\)

hc tốt

a) a² + b² + c2 = ab + ac + bc

=> 2a² + 2b² + 2c² = 2ab + 2ac + 2bc

=> 2a² + 2b² + 2c² - 2ab - 2ac - 2bc = 0

=> (a² - 2ab + b²) + (a² - 2ac + c²) + (b² - 2bc + c²) = 0

=> (a - b)² + (a - c)² + (b - c)² = 0 

Do 3 hạng tử trên đều có giá trị lớn hơn hoặc bằng 0 nên a - b = a - c = b - c = 0

=> a = b = c 

b) a³ + b³ + c³ = 3abc

=> a³ + b³ + c³ - 3abc = 0

=> a³ + 3a²b + 3ab² + b³ + c³ - 3abc - 3a²b - 3ab² = 0

=> (a + b)³ + c³ - 3ab(a + b + c) = 0

=> (a + b + c)(a² + 2ab + b² - bc - ac + c²) - 3ab(a + b + c) = 0

=> (a + b + c)(a² + b² + c² - ab - bc - ac) = 0 

=> a + b + c = 0

hoặc a² + b² + c² = ab + bc + ac =>  a = b = c

a)\(a^2+b^2+c^2=ab+bc+ca\)\(\Rightarrow2a^2+2b^2+2c^2=2ab+2bc+2ca\)

\(\Rightarrow2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)\(\Rightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ca+a^2\right)=0\)

\(\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)

\(\Rightarrow\hept{\begin{cases}a=b\\b=c\\c=a\end{cases}\Rightarrow a=b=c}\)

b)\(a^3+b^3+c^3=3abc\Rightarrow a^3+b^3+c^3-3abc=0\)

\(\Rightarrow\left(a+b\right)^3+c^3-3ab\left(a+b+c\right)=0\)

\(\Rightarrow\left(a+b+c\right)\left(a^2+2ab+b^2-ac-bc+c^2\right)-3ab\left(a+b+c\right)=0\)

\(\Rightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0\)

\(\Rightarrow\hept{\begin{cases}a+b+c=0\\a=b=c\end{cases}}\)

Đặt \(\frac{\left(a+b-c\right)}{2}=x;\frac{\left(c+a-b\right)}{2}=y;\frac{\left(b+c-a\right)}{2}=z\) thì x, y, z > 0(do a, b, c là độ dài 3 cạnh tam giác)

Và \(a=x+y;b=x+z;c=y+z\)

Thay vào, ta cần chứng minh: \(2\left[xy\left(x+y\right)+yz\left(y+z\right)+zx\left(z+x\right)+6xyz\right]>0\) (luôn đúng do x, y, z > 0)

Done!

Do \(c^2+2\left(ab-ac-bc\right)=0\Leftrightarrow-c^2=2\left(ab-ac-bc\right)\) 

Ta có; \(\frac{a^2+\left(a-c\right)^2}{b^2+\left(b-c\right)^2}=\frac{a^2+c^2-c^2+\left(a-c\right)^2}{b^2+c^2-c^2+\left(b-c\right)^2}=\frac{a^2+c^2+2\left(ab-ac-bc\right)+\left(a-c\right)^2}{b^2+c^2+2\left(ab-ac-bc\right)+\left(b-c\right)^2}\)

\(=\frac{2\left(a-c\right)^2+2\left(ab-bc\right)}{2\left(b-c\right)^2+2\left(ab-ac\right)}=\frac{2\left(a-c\right)^2+2b\left(a-c\right)}{2\left(b-c\right)^2+2a\left(b-c\right)}=\frac{\left(a-c\right)\left(a-c+b\right)}{\left(b-c\right)\left(b-c+a\right)}\)

\(=\frac{a-c}{b-c}\) (đpcm)