Does exist a triangle with sides a integer length where one of height is equal to the side which is the base? Does exist a triangle with sides a integer length where one of height is equal to the side which is the base?

$a,k,l$ -natural,
$a$ is lenght of hight

In formulas, we want to find positive integers $a,b,c,d$ solving the Diophantine equations
$$
a^2+(a+b)^2=c^2
$$
$$
b^2+(a+b)^2=d^2.
$$
In particular, we obtain the Diophantine equation
$$
a^2+d^2=b^2+c^2,
$$
which has been studied [here](
Diophantine equation $a^2+b^2=c^2+d^2$). hence there are integers $p,q,r,s$ such that
$$
(a,b,c,d)=(pr+qs,ps+qr,pr-qs,qr-ps).
$$
Use this to show that there are no solutions (if I am not wrong); the equations then are given by
$$
(2pr + ps + qr)(ps + qr + 2qs) + (pr + qs)^2=0,
$$
$$
(pr + 2ps + qs)(pr + 2qr + qs) + (ps + qr)^2=0.
$$
