If $p$ is prime and $p$ divides $a^2+b^2 $and $p$ divides $c^2+d^2$ does $p$ divide $a^2-c^2$?

Question

I need some help getting started on proving this. If $p$ is prime and $p$ divides $a^2+b^2$ and $p$ divides $c^2+d^2$ does $p$ divide $a^2-c^2$? Thanks in advanced!

Answer 1

Let $p=13$, $a=4$, $b=7$, $c=1$, $d=8$.

There are simpler examples such as $p=5$, $a=5$, $b=0$, $c=3$, $d=4$, but perhaps you wanted $a,b,c,d$ to be positive integers.