Question

Cho :\(a+b+c=a^2+b^2+c^2=1\)Và \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\) Với (a,b,c#0)

Chứng minh rằng: \(\left(x+y+z\right)^2=x^2+y^2+z^2\)

Answer 1

1=x+y+z
<=> x = a(x+y+z) ; y = b(x+y+z) ; z = c(x+y+z) 
=> xy+yz+zx=  ab(x+y+z)2+bc(x+y+z)2+ca(x + y + z)2
<=> xy+yz+zx =(ab+bc+ca)(x+y+z)2 (2) 
từ (1) và (2) => xy + yz + zx = 0

Answer 2

Mình nhâm sorry

xa=yb=zc" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline; float:none; line-height:normal; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax">xa=yb=zc

ax=by=cz" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline; float:none; line-height:normal; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax">ax=by=cz

ax=by=cz=a+b+cx+y+z=1x+y+z" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline; float:none; line-height:normal; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax">ax=by=cz=a+b+cx+y+z=1x+y+z (Vì a + b + c = 1) (*1)

ax=by=cz" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline; float:none; line-height:normal; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax">ax=by=cz

ax)2=(by)2=(cy)2" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline; float:none; line-height:normal; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax">ax)2=(by)2=(cy)2= a2x2=b2y2=c2z2" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline; float:none; line-height:normal; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax">a2x2=b2y2=c2z2

a2x2=b2y2=c2z2=a2+b2+c2x2+y2+z2=1x2+y2+z2" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline; float:none; line-height:normal; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax">a2x2=b2y2=c2z2=a2+b2+c2x2+y2+z2=1x2+y2+z2 (*2)

1x+y+z)2=1x2+y2+z2" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline; float:none; line-height:normal; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax">1x+y+z)2=1x2+y2+z2

12(x+y+z)2=1x2+y2+z2" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline; float:none; line-height:normal; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax">12(x+y+z)2=1x2+y2+z2

1(x+y+z)2=1x2+y2+z2" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline; float:none; line-height:normal; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax">1(x+y+z)2=1x2+y2+z2

=> (x+y+z)2=x2+y2+z2(x+y+z)2=x2+y2+z2 (ĐPCM) (Vì hai phân số bằng nhau,tử số bằng nhau => mẫu số bằng nhau.)