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    "# [Right Triangles with Integer Coordinates](https://projecteuler.net/problem=91)\n",
    "\n",
    "We'll start by generating all the possible values of $P$ and $Q$."
   ]
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    "limit = 50\n",
    "points = ((x, y) for x in range(0, limit + 1) for y in range(0, limit + 1) if (x, y) != (0, 0))"
   ]
  },
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   "source": [
    "If we think about $P$ and $Q$ as vectors instead of points, we can solve this problem with [dot products](https://en.wikipedia.org/wiki/Dot_product). 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 $\\vec{Q} \\cdot \\vec{Q}$."
   ]
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    {
     "data": {
      "text/plain": [
       "14234"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "from itertools import combinations\n",
    "\n",
    "triangles = set()\n",
    "for ((x1, y1), (x2, y2)) in combinations(points, 2):\n",
    "    d = x1 * x2 + y1 * y2\n",
    "    if d == 0 or d == x1^2 + y1^2 or d == x2^2 + y2^2:\n",
    "        triangles.add(((x1, y1), (x2, y2)))\n",
    "        \n",
    "len(triangles)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35a4a234",
   "metadata": {},
   "source": [
    "## Relevant sequences\n",
    "* Answers for limits of 0, 1, 2, ...: [A155154](https://oeis.org/A155154)\n",
    "\n",
    "#### Copyright (C) 2025 filifa\n",
    "\n",
    "This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
   ]
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