## Biclinic Integral Quadrilaterals

### Problem 311

Published on Saturday, 20th November 2010, 10:00 pm; Solved by 302; Difficulty rating: 70%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 AB^{2}+BC^{2}+CD^{2}+AD^{2} ≤ `N`.

We can verify that B(10 000) = 49 and B(1 000 000) = 38239.

Find B(10 000 000 000).