--description--

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?

--hints--

threeSimilarTriangles() should return 549936643.

assert.strictEqual(threeSimilarTriangles(), 549936643);

--seed--

--seed-contents--

function threeSimilarTriangles() {

  return true;
}

threeSimilarTriangles();

--solutions--

// solution required