X, Y are inner-product space A is a mapping from X to Y, and ||Ax-Ay||=||x-y|| and A0=0, how to prove A is a linear operator??

Question

It seems not easy for proving.

As we know, ||Ax||_Y=||x||_X, then A preserves the distance, but how to use this condition to prove A is a linear operator? I feel confused.

Thank you very much!