sol1.py

"""
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?
"""

fromitertoolsimportcombinations, product


defis_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
 """
ifx1==y1==0orx2==y2==0:
returnFalse
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
ora_square+c_square==b_square
orb_square+c_square==a_square
 )


defsolution(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
 """
returnsum(
1
forpt1, pt2incombinations(product(range(limit+1), repeat=2), 2)
ifis_right(*pt1, *pt2)
 )


if__name__=="__main__":
print(f"{solution() =}")