Right Triangles with Integer Coordinates

We'll start by generating all the possible values of $P$ and $Q$.

limit = 50
points = ((x, y) for x in range(0, limit + 1) for y in range(0, limit + 1) if (x, y) != (0, 0))

If we think about $P$ and $Q$ as vectors instead of points, we can solve this problem with dot products. Since the dot product of orthogonal vectors is 0, we can check for a right angle in the triangle by seeing if $\vec{P} \cdot \vec{Q} = 0$, $\vec{P} \cdot (\vec{Q} - \vec{P}) = 0$, or $\vec{Q} \cdot (\vec{Q} - \vec{P}) = 0$. By distributing in the last two equations, we can simply check if $\vec{P} \cdot \vec{Q}$ equals 0, $\vec{P} \cdot \vec{P}$, or $\vec{Q} \cdot \vec{Q}$.

from itertools import combinations

triangles = set()
for ((x1, y1), (x2, y2)) in combinations(points, 2):
    d = x1 * x2 + y1 * y2
    if d == 0 or d == x1^2 + y1^2 or d == x2^2 + y2^2:
        triangles.add(((x1, y1), (x2, y2)))
        
len(triangles)

14234

Relevant sequences

• Answers for limits of 0, 1, 2, ...: A155154

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