Biclinic Integral Quadrilaterals

$A B C D$ is a convex, integer sided quadrilateral with $1 \leq A B < B C < C D < A D$ .
$B D$ has integer length. $O$ is the midpoint of $B D$ . $A O$ has integer length.
We'll call $A B C D$ a biclinic integral quadrilateral if $A O = C O \leq B O = D O$ .

For example, the following quadrilateral is a biclinic integral quadrilateral:
$A B = 19$ , $B C = 29$ , $C D = 37$ , $A D = 43$ , $B D = 48$ and $A O = C O = 23$ .

Let $B (N)$ be the number of distinct biclinic integral quadrilaterals $A B C D$ that satisfy $A B^{2} + B C^{2} + C D^{2} + A D^{2} \leq N$ .
We can verify that $B (10 000) = 49$ and $B (1 000 000) = 38239$ .

Find $B (10 000 000 000)$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.