Three Similar Triangles

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

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 $b + d \lt 100$, there are $92$ distinct triplets $(a, b, d)$ such that point $P$ exists.
If $b + d \lt 100\,000$, there are $320471$ distinct triplets $(a, b, d)$ such that point $P$ exists.
If $b + d \lt 100\,000\,000$, how many distinct triplets $(a, b, d)$ are there such that point $P$ exists?

To solve this problem, we can iterate through all possible values of $a$, $b$, and $d$, and check if the conditions for point $P$ exist. This can be done using nested for loops.

Here is the Python code to solve the problem:

“`python
count = 0

# Iterate through all possible values of a
for a in range(1, 1001):
# Iterate through all possible values of b
for b in range(a+1, 1001):
# Iterate through all possible values of d
for d in range(1, 1001):
# Check if the condition a = c is satisfied
if a == d:
# Check if the condition for point P exists
if (b + d) < 100000000: count += 1 print(count) ``` This code will iterate through values of $a$ from 1 to 1000, $b$ from $a+1$ to 1000, and $d$ from 1 to 1000. It checks if $a = d$ and if the condition $b + d < 100000000$ is satisfied. If both conditions are met, it increments the count variable. The output of this code will be the number of distinct triplets $(a, b, d)$ that satisfy the given conditions for point $P$.

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