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Problem 299

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Three similar triangles

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.

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?

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三个相似三角形

给定四个坐标为整数的点：A(a, 0)、B(b, 0)、C(0, c)和D(0, d)，其中0 < a < b且0 < c < d。
在直线AC上选择另一个坐标为整数的点P，使得三角形ABP、CDP和BDP均相似。

很容易证明仅当a=c时这三个三角形可能相似。

因此，给定a=c，我们需要找到三元组(a,b,d)使得AC上至少存在一个坐标为整数的点P满足三角形ABP、CDP和BDP均相似。

例如，如果(a,b,d)=(2,3,4)，很容易验证点P(1,1)满足上述条件。注意三元组(2,3,4)和(2,4,3)被认为是不同的三元组，尽管P(1,1)这两个三元组来说都满足条件。

如果要求b+d < 100，一共有92个不同的三元组(a,b,d)使得点P存在。
如果要求b+d < 100 000，一共有320471个不同的三元组(a,b,d)使得点P存在。

如果要求b+d < 100 000 000，一共有多少个不同的三元组(a,b,d)使得点P存在？

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