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puzzles/project_euler/pe/p143.html
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puzzles/project_euler/pe/p143.html
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<meta charset="utf-8" />
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<meta name="author" content="Colin Hughes" />
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<meta name="description" content="A website dedicated to the fascinating world of mathematics and programming" />
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<meta name="keywords" content="programming,mathematics,problems,puzzles" />
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<title>Problem 143 - Project Euler</title>
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<h2>Investigating the Torricelli point of a triangle</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 143</h3><span style="width:300px;color:#666;">Published on Friday, 2nd March 2007, 09:00 pm; Solved by 1270</span></div>
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<div class="problem_content" role="problem">
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<p>Let ABC be a triangle with all interior angles being less than 120 degrees. Let X be any point inside the triangle and let XA = p, XC = q, and XB = r.</p>
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<p>Fermat challenged Torricelli to find the position of X such that p + q + r was minimised.</p>
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<p>Torricelli was able to prove that if equilateral triangles AOB, BNC and AMC are constructed on each side of triangle ABC, the circumscribed circles of AOB, BNC, and AMC will intersect at a single point, T, inside the triangle. Moreover he proved that T, called the Torricelli/Fermat point, minimises p + q + r. Even more remarkable, it can be shown that when the sum is minimised, AN = BM = CO = p + q + r and that AN, BM and CO also intersect at T.</p>
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<div style='text-align:center;'><img src='project/images/p_143_torricelli.gif' width='564' height='560' alt='' /></div>
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<p>If the sum is minimised and a, b, c, p, q and r are all positive integers we shall call triangle ABC a Torricelli triangle. For example, a = 399, b = 455, c = 511 is an example of a Torricelli triangle, with p + q + r = 784.</p>
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<p>Find the sum of all distinct values of p + q + r <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 120000 for Torricelli triangles.</p>
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