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Three similar triangles

Problem 299

Published on Saturday, 3rd July 2010, 01:00 am; Solved by 424; Difficulty rating: 60%

Four points with integer coordinates are selected:
A(a, 0), B(b, 0), C(0, c) and D(0, d), with 0 < a < b and 0 < c < d.
Point P, also with integer coordinates, is chosen on the line AC so that the three triangles ABP, CDP and BDP are all similar.

p299_ThreeSimTri.gif

It is easy to prove that the three triangles can be similar, only if a=c.

So, given that a=c, we are looking for triplets (a,b,d) such that at least one point P (with integer coordinates) exists on AC, making the three triangles ABP, CDP and BDP all similar.

For example, if (a,b,d)=(2,3,4), it can be easily verified that point P(1,1) satisfies the above condition. Note that the triplets (2,3,4) and (2,4,3) are considered as distinct, although point P(1,1) is common for both.

If b+d < 100, there are 92 distinct triplets (a,b,d) such that point P exists.
If b+d < 100 000, there are 320471 distinct triplets (a,b,d) such that point P exists.

If b+d < 100 000 000, how many distinct triplets (a,b,d) are there such that point P exists?