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"""
Project Euler Problem 91: https://projecteuler.net/problem=91
The points P (x1, y1) and Q (x2, y2) are plotted at integer coordinates and
are joined to the origin, O(0,0), to form ΔOPQ.

There are exactly fourteen triangles containing a right angle that can be formed
when each coordinate lies between 0 and 2 inclusive; that is,
0 ≤ x1, y1, x2, y2 ≤ 2.

Given that 0 ≤ x1, y1, x2, y2 ≤ 50, how many right triangles can be formed?
"""
from itertools import combinations, product
def is_right(x1: int, y1: int, x2: int, y2: int) -> bool:
"""
Check if the triangle described by P(x1,y1), Q(x2,y2) and O(0,0) is right-angled.
Note: this doesn't check if P and Q are equal, but that's handled by the use of
itertools.combinations in the solution function.
>>> is_right(0, 1, 2, 0)
True
>>> is_right(1, 0, 2, 2)
False
"""
if x1 == y1 == 0 or x2 == y2 == 0:
return False
a_square = x1 * x1 + y1 * y1
b_square = x2 * x2 + y2 * y2
c_square = (x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2)
return (
a_square + b_square == c_square
or a_square + c_square == b_square
or b_square + c_square == a_square
)
def solution(limit: int = 50) -> int:
"""
Return the number of right triangles OPQ that can be formed by two points P, Q
which have both x- and y- coordinates between 0 and limit inclusive.
>>> solution(2)
14
>>> solution(10)
448
"""
return sum(
1
for pt1, pt2 in combinations(product(range(limit + 1), repeat=2), 2)
if is_right(*pt1, *pt2)
)
if __name__ == "__main__":
print(f"{solution() = }")
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