Question

If x= b+c, y= c+a,z=a+b then prove that x^2+y^2+z^2 - a^2 - b^2 - c^2=xy+yz+zx -ab - bc -ca

Answer 1

x=b+c..................................................1
squaring on both sides we get,
x^2=b^2+c^2+2bc...........................2
y=c+a..................................................3
squaring on both sides we get,
y^2=c^2+a^2+2ac............................4
z=a+b...................................................5
squaring on both sides we get,
z^2=a^2+b^2+2ab.............................6
multiplying eq1 and eq3,we get
xy=bc+ab+ac+c^2.............................7
multiplying eq3 and eq5 we get
yz=ac+bc+ab+a^2.............................8
multiplying eq1 and eq5 we get
xz=ab+ac+bc+b^2..............................9
By adding eq2,eq4,eq6, we get
x^2+y^2+z^2=b^2+c^2+2bc+c^2+a^2+2ac
a^2+b^2++2ab
x^2+y^2+z^2-a^2-b^2-c^2=a^2+b^2+c^2+2ab+2bc+2ac
Adding ab+bc+ac on both sides we get
x^2+y^2+z^2-a^2-b^2-c^2+ab+bc+ac=a^2+b^2+c^2+ab+bc+ac+ab+bc+ac+ab+bc+ac
By eq7,8,9 we get
x^2+y^2+z^2-a^2-b^2-c^2+ab+bc+ac=xy+yz+xz
x^2+y^2+z^2-a^2-b^2-c^2=xy+yz+xz-ab-bc-ac
Hence proved.