## Biclinic Integral Quadrilaterals

### Problem 311

Published on Saturday, 20th November 2010, 10:00 pm; Solved by 296

ABCD is a convex, integer sided quadrilateral with 1 AB BC CD AD.
BD has integer length. O is the midpoint of BD. AO has integer length.
We'll call ABCD a biclinic integral quadrilateral if AO = CO BO = DO.

For example, the following quadrilateral is a biclinic integral quadrilateral:
AB = 19, BC = 29, CD = 37, AD = 43, BD = 48 and AO = CO = 23.

Let B(N) be the number of distinct biclinic integral quadrilaterals ABCD that satisfy AB2+BC2+CD2+AD2 N.
We can verify that B(10 000) = 49 and B(1 000 000) = 38239.

Find B(10 000 000 000).