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Square on the Inside

Problem 504 Published on Saturday, 21st February 2015, 10:00 pm; Solved by 2309;
Difficulty rating: 15%

Let ABCD be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows:

A(a, 0), B(0, b), C(−c, 0), D(0, −d), where 1 ≤ a, b, c, dm and a, b, c, d, m are integers.

It can be shown that for m = 4 there are exactly 256 valid ways to construct ABCD. Of these 256 quadrilaterals, 42 of them strictly contain a square number of lattice points.

How many quadrilaterals ABCD strictly contain a square number of lattice points for m = 100?