If x = a + m, y = b + m, z = c + m, then the value of $$\frac{{{x^2} + {y^2} + {z^2} - yz - zx - xy}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}}$$       is = ?

Solution (By Examveda Team)

$$\eqalign{ & \frac{{{x^2} + {y^2} + {z^2} - yz - zx - xy}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}}{\text{ }} \cr & = \frac{{{{\left( {a + m} \right)}^2} + {{\left( {b + m} \right)}^2} + {{\left( {c + m} \right)}^2} - \left( {b + m} \right)\left( {c + m} \right) - \left( {c + m} \right)\left( {a + m} \right) - \left( {a + m} \right)\left( {b + m} \right)}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}} \cr & = \frac{{{a^2} + {m^2} + 2am + {b^2} + {m^2} + 2bm + {c^2} + {m^2} + 2cm - bc - bm - cm - {m^2} - ca - cm - am - {m^2} - ab - am - bm - {m^2}}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}} \cr & = \frac{{{a^2} + {b^2} + {c^2} - ab - bc - ca}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}} \cr & = 1 \cr} $$