Multiplying the given equality by $(x-a)(x-b)$, we get $(x-a)^2 = (x-b)^2$. Squaring both sides and subtracting $x^2$ we get $-2a x + a^2 = -2b x + b^2$, which can be further rearranged into $2(b-a)x = (b-a)(b+a)$. Since $b-a \neq 0$, we can divide both sides of the equality by $b-a$ to get $2x = b+a$. From here we get $x = \frac{a+b}{2}$.