From 0fcba4a2e6557df0456aef7d57529dd20ae69fd0 Mon Sep 17 00:00:00 2001
From: filifa <filifa.breath652@8alias.com>
Date: Tue, 17 Jun 2025 23:08:40 -0400
Subject: [PATCH] add problem 91

---
 notebooks/problem0091.ipynb | 92 +++++++++++++++++++++++++++++++++++++
 1 file changed, 92 insertions(+)
 create mode 100644 notebooks/problem0091.ipynb

diff --git a/notebooks/problem0091.ipynb b/notebooks/problem0091.ipynb
new file mode 100644
index 0000000..4c2aeaa
--- /dev/null
+++ b/notebooks/problem0091.ipynb
@@ -0,0 +1,92 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "dcfe1d77",
+   "metadata": {},
+   "source": [
+    "# [Right Triangles with Integer Coordinates](https://projecteuler.net/problem=91)\n",
+    "\n",
+    "We'll start by generating all the possible values of $P$ and $Q$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "cba05ca7",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "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))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "90cb43dd",
+   "metadata": {},
+   "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 $\\vec{Q} \\cdot \\vec{Q}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "18e334eb",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "14234"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "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)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "SageMath 9.5",
+   "language": "sage",
+   "name": "sagemath"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}