Is the proportionality characteristic of this function being carried on?

$$A=kx$$ is a directly proportional function,where $$A^2=B^2+C^2$$.Does it necessarily mean $$B$$ and $$C$$ both vary directly with respect to $$x$$? If not, under what condition is this possible?

No. If $$B(x)$$ is an arbitrary function bounded between $$0$$ and $$\sqrt{kx}$$, we can simply let $$C(x) = \sqrt{kx - B(x)^2}$$, and the conditions are satisfied. Your condition will need to restrict $$B(x)$$ to be linear if you wish for $$B$$ and $$C$$ to be proportional to $$x$$.