98 lines
5.6 KiB
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98 lines
5.6 KiB
HTML
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<h2>Triangle Centres</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 264</h3><span style="width:300px;color:#666;">Published on Saturday, 14th November 2009, 05:00 am; Solved by 400</span></div>
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<p>Consider all the triangles having:
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<ul>
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<li>All their vertices on <dfn title="Integer coordinates">lattice points</dfn>.</li>
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<li><dfn title="Centre of the circumscribed circle">Circumcentre</dfn> at the origin O.</li>
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<li><dfn title="Point where the three altitudes meet">Orthocentre</dfn> at the point H(5, 0).</li>
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</ul></p>
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<p>There are nine such triangles having a perimeter <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 50.<br />
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Listed and shown in ascending order of their perimeter, they are:</p>
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<p><table>
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<tr>
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<td>A(-4, 3), B(5, 0), C(4, -3)<br />
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A(4, 3), B(5, 0), C(-4, -3)<br />
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A(-3, 4), B(5, 0), C(3, -4)<br /><br /><br />
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A(3, 4), B(5, 0), C(-3, -4)<br />
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A(0, 5), B(5, 0), C(0, -5)<br />
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A(1, 8), B(8, -1), C(-4, -7)<br /><br /><br />
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A(8, 1), B(1, -8), C(-4, 7)<br />
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A(2, 9), B(9, -2), C(-6, -7)<br />
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A(9, 2), B(2, -9), C(-6, 7)<br /></td>
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<td><img src="project/images/p_264_TriangleCentres.gif" /></td>
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</tr>
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</table></p>
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<p>The sum of their perimeters, rounded to four decimal places, is 291.0089.</p>
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<p>Find all such triangles with a perimeter <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 10<sup>5</sup>.<br />
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Enter as your answer the sum of their perimeters rounded to four decimal places.</p>
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