Question

Cho (a - b)² + (b - c)² + (c - a)² = 4(a² +b² + c² - ab - bc - ca). Chứng minh rằng  a = b = c

Answer 1

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

\(\Leftrightarrow2a^2-2ab-2bc-2ca+2b^2+2c^2=4a^2+4b^2+4c^2-4ab-4bc-2ca\)

\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)

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

Dấu ''='' xảy ra khi a = b = c