Do there exist postive integers $a,b,c,x,y,p,q$ such $(a,b,c)$, $(x,y,a)$, $(p,q,b)$ are all Pythagorean triples? That is, does the system
$$\begin{cases}
a^2+b^2=c^2\\
x^2+y^2=a^2\\
p^2+q^2=b^2
\end{cases}$$
have a postive integer solution?



Is it possible that $(a,b,c)$, $(x,y,a)$, $(p,q,b)$ are Pythagorean triples simultaneously?
Yes, for example 
$$ (a,b,c,x,y,p,q) = (145,10512,10513,143,24,7920,6912).$$


**P.S.** I found this example by looking for $(u,v)$ such that $u,v,u+v,u-v$ are all sums of two squares (cf. ARupinski's comment). I took $u=73$ and $v=72$.


