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project_euler/pe/p138.html
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project_euler/pe/p138.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 138 - Project Euler</title>
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<h2>Special isosceles triangles</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 138</h3><span style="width:300px;color:#666;">Published on Saturday, 20th January 2007, 11:00 am; Solved by 2893</span></div>
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<div class="problem_content" role="problem">
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<p>Consider the isosceles triangle with base length, <i>b</i> = 16, and legs, L = 17.</p>
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<div style='text-align:center;'>
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<img src='project/images/p_138.gif' width='230' height='228' alt='' />
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</div>
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<p>By using the Pythagorean theorem it can be seen that the height of the triangle, <i>h</i> = <img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />(17<sup>2</sup> <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> 8<sup>2</sup>) = 15, which is one less than the base length.</p>
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<p>With <i>b</i> = 272 and L = 305, we get <i>h</i> = 273, which is one more than the base length, and this is the second smallest isosceles triangle with the property that <i>h</i> = <i>b</i> <img src='images/symbol_plusmn.gif' width='11' height='11' alt='±' border='0' style='vertical-align:middle;' /> 1.</p>
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<p>Find <img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' /> L for the twelve smallest isosceles triangles for which <i>h</i> = <i>b</i> <img src='images/symbol_plusmn.gif' width='11' height='11' alt='±' border='0' style='vertical-align:middle;' /> 1 and <i>b</i>, L are positive integers.</p>
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