Problem 57954. Easy Sequences 108: Enclosing a Circle with Primitive Pythagorean Triangles

Created by Ramon Villamangca

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Pythagorean_triple\"><w:r><w:rPr><w:b/></w:rPr><w:t>Pythagorean Triangle</w:t></w:r></w:hyperlink><w:r><w:t> is a right triangle all sides of which are represented by integers. Examples are triangles with sides </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>[3\\ 4\\ 5]</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>[5\\ 12\\ 13]</w:t></w:r></w:customXml><w:r><w:t>. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>We say that pythagorean triangle is a </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>primitive pythagorean triangle (PPT), </w:t></w:r><w:r><w:t>if all of its sides are relatively prime. That is, if the sides of a pythagorean triangle are </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:t>, </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b</w:t></w:r><w:r><w:t> and </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>c</w:t></w:r><w:r><w:t>, it is PPT if and only if: </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>gcd(a,b) == 1 &amp;&amp; gcd(a,c) == 1 &amp;&amp; gcd(b,c) == 1</w:t></w:r><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Given a circle of radius </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R</w:t></w:r><w:r><w:t>, find the dimensions of the </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>PPT with the smallest area, that can enclose the circle, without touching the circle </w:t></w:r><w:r><w:t>(i.e. no sides is a tangent of the circle, or crosses the circle).</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t> </w:t></w:r><w:r><w:t>For example, in the figure below, if</w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t> R = 12</w:t></w:r><w:r><w:t>, the smallest pythagorean triangle that can enclose the circle without touching it is </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>[39,52,65]</w:t></w:r><w:r><w:t>. However this is not primitive since, </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>gcd(39,52) = 13 ≠ 1</w:t></w:r><w:r><w:t>. The correct smallest PPT is the triangle with dimensions: </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>[48 55 73]</w:t></w:r><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> </w:t></w:r><w:customXml w:element=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"202\"/><w:attr w:name=\"width\" w:val=\"354\"/><w:attr w:name=\"verticalAlign\" w:val=\"baseline\"/><w:attr w:name=\"altText\" w:val=\"\"/><w:attr w:name=\"relationshipId\" w:val=\"rId1\"/></w:customXmlPr></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>-----------------------</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>NOTE: </w:t></w:r><w:r><w:t> There can be more than one PPT with smallest area for the same value of </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>. </w:t></w:r><w:r><w:t>For example if </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R = 4.5</w:t></w:r><w:r><w:t>, there are two possible smallest area PPT: </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>[20 21 29] </w:t></w:r><w:r><w:t>and </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>[12 35 37]</w:t></w:r><w:r><w:t>. Both have an area of </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>210</w:t></w:r><w:r><w:t>. </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>In cases where there are more than one possible PPT's, the function should output the PPT with the least perimeter.</w:t></w:r><w:r><w:t> So, for </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R = 4.5</w:t></w:r><w:r><w:t>, the function output should be: </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>[20 21 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<div style = "text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; "><div style="block-size: 547px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 273.5px; transform-origin: 407px 273.5px; vertical-align: baseline; "><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><a target='_blank' href = "https://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean Triangle</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "> is a right triangle all sides of which are represented by integers. Examples are triangles with sides <img src="data:image/png;base64,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" width="50" height="19" style="width: 50px; height: 19px;"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "> and <img src="data:image/png;base64,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" width="65" height="19" style="width: 65px; height: 19px;"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">. </div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">We say that pythagorean triangle is a <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">primitive pythagorean triangle (PPT), <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">if all of its sides are relatively prime. That is, if the sides of a pythagorean triangle are <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">a<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">, <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">b<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "> and <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">c<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">, it is PPT if and only if: <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">gcd(a,b) == 1 &amp;&amp; gcd(a,c) == 1 &amp;&amp; gcd(b,c) == 1<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">.</div><div style="block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">Given a circle of radius <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">R<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">, find the dimensions of the <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">PPT with the smallest area, that can enclose the circle, without touching the circle <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">(i.e. no sides is a tangent of the circle, or crosses the circle).<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">For example, in the figure below, if<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "> R = 12<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">, the smallest pythagorean triangle that can enclose the circle without touching it is <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">[39,52,65]<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">. However this is not primitive since, <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">gcd(39,52) = 13 ≠ 1<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">. The correct smallest PPT is the triangle with dimensions: <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">[48 55 73]<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">.</div><div style="block-size: 208px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 104px; text-align: left; transform-origin: 384px 104px; white-space: 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MKwPK/se7FqwxZzOfsz9kJ58s+nfb7cdhnfqK5tory2e3uUDxyDDKf8APX3rKrTVSNuphKPMjD1bU49Lt4ZJMEyzpEoP+03P5DJpXf8AtKDFjdbIxIVkkReSB1Cnp14zz371znirw/qV3dWsN7domnxZC3bLk5OMbwOh7Z4B9s4re0jSpdLt1ik1Ce6VVChZAoUfTjP615TUlJxkjm1vZmlRRRVDCiiigAooooAKpmI6vdm0QZs4WH2p+zkc+UP03e3HfgR59VYxac3l24OJLzH5iP1Pv0Hv0ratbaGztkt7ZAkaDAH9fc+9dFGi5vmexcY31ZLRRRXom4Vxfjb/AJGvwl/1+/8As8VdpXF+Nv8Aka/CX/X7/wCzxVvh/wCJ8n+TOfEfw/mvzR3E80dtbyTzuI4olLu7dFUDJNeGeLxNrugzeLbsMou9QW2skP8ABAqSfzI/MGvatX0uHWtIuNOunkSG4TY7REBgPbINed+K/hpdr4ft7Xw9e6pqBjnXFpd3ieVGm1vmVSFAIJA47E1z4GpTpyvJ2d/wOrFQnONkrr9Sz8WIjDLoupXAjubG1nPnWTyBfN5B4B+9wCOhx+dUvAjWer/Ea+1TSIY9LtIoNn2EEK7kgAnYOAM/ritTxJ4Q1aPVNE1HSN2s/wBmKVaG/nHmSfMW3FjgE849sDrT9J8P61qHxETxNqunQ6RHDEU+zpOsrSkqVyxXg/e6+wrWNSCw/LzdH189rfqZuMnW5rdV/wAPf9DvaKKK8g9EKKKKAOY8UeA9J8Tfv3U2l+vKXkAw2R03D+L+foRWHaeMPFHw/mSz8ZW76rpOQsepQ8ug7ZJ6/RsH3Neh0yWGOeF4p41kjcbWR1yGHoQazdOz5oOz/rc9SjmDVNUMTH2lPs91/he6/LyLmj65puv6et7o93HdQN3Q8qfQjqD7Gr9eT6l8PrzR9QbWPh/fNp12OXtC37qUenPH4HI+lanhz4qwy3g0jxnbHRdUXClpARDIfXJ+7n34962hideWro/wZnWylVIOtgJc8Vuvtx9V1XmvwPRKKQEMoZSCCMgjvS11nghRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAVxHxP0nX/EGjWukaBbb4biYG8lMqoEQYwCCQSMnPGfu129FXTm4SUl0JlHmVmZkenzaP4YSw0BIfOtbcR2wuM7CwHG7HPPeuD8U6f438b6ZDo134ftNLhEyyS3bXqyg4zyqjkdfevT6KuFVwfNa7FKCkrdCtp1munaXa2SOXW2hSIMRywVQM/pVmiism7u5YVyfin4f6d4gc3lqx0/U1O5bmEY3N23AdfqOa6yiqhUlTfNF2InCNRcskeQS69rHhvU7TT/GkWUR8xX0YyJFBGTx1xx788iu7trmC8t0ntZUmicZV0OQa5D4uQpceK/BMEy7opr145F/vKXhBH5VFd+F9Z8J3D33hCZri1J3S2Eh3ce3r/P61pCtSrzlB+7Jfc9Pw/I6K+ErYXDUa1+eM035q0mvntfudzRXPeHvGNhr37g5tL5eGtpTg577T3/n7V0NOcJQdpI5oTjNXixGVXQo6hlYYIIyCKyH0WW1O7SJxGn/PrNlo/wDgJ6p+o9q2KKynCM17yKcU9zAe9ktuNQsri3/21TzU/wC+lzj8QKfDqNlccQXcEh9FkBP5VuVBPZWt0MXVtDN/10jDfzrllhezM3T7FLcuM5GPXNV5tRsrf/X3cMZ9GkAP5Vc/sDR92f7Jsc+v2ZP8KswWVra/8ettDD/1zjC/yqfqsu4vZsx0vpLk40+yuLj/AG2TykH/AAJsZ/AGp00aa6IbV5w6f8+sGVj/AOBHq/6D2rYoraOHhHV6lKCW4iIsaKkahVUYCgYAFLRRXSaBRRRQAVxfjb/ka/CX/X7/AOzxV2lcX42/5Gvwl/1+/wDs8Vb4f+J8n+TOfEfw/mvzR3dFFFeKeuFFFFABRRRQAUUUUAFFFFABWXrvhvS/Eln9n1a1WUD7kg4eM+qt2/lWpRSaTVmXTqTpTU6bs11R5tFF4u+GbbtNdtf8PqctbPnzIV9u4+oyPUCu/wDCvjjRfF9tu0y423CjMlrL8sifh3HuMirVcd4k+HVjq1z/AGjpEraRqyHelxb/AChm9SB39xz9aiPtKXwart/kz1pV8LmGmLXJU/nit/8AFHr6rX1PSaK8q0z4jaz4UvI9L+Itm3lk7YtUgXKuPVgOv4YPqK9Nsb+11Ozju9PuI7m3kGUkiYMprrp1oVNt+3U8jGZdXwdnNXi9pLWL9H+m5YooorY88KKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooA8n+M17HpviDwdfTqzRWt1JM4QAsVV4ScZ78VH/wurw7/wA+Wqf9+o//AIuvXKK45Yebm5xla/l/wT36WZ4X6tToV6Dk4X1U7btvblffueA+I/G3gzxB+/FlqlpfLylzFDGDntuHmc/z96s+Fvig8SrbawXuoRws+P3qj/aH8X8/rXutQ3dpb31q9teQxzwyDDxyLkH8K7sPUqwXJVlzR9NV6O55mMlgasebDUXTn357p+q5Vf70zkrfxFY3cCzWxaWNujLg/wBak/tm3/uS/kP8axNT8C6r4Zun1PwNOzRE7pdOlO4MPbP3v5+hNW/D3jay1mT7HdqbDUVO1rabjLd9pP8AI811Ojdc9N3X4r1POjiIJ8lSFn66P0ND+2bf+5L+Q/xo/tm3/uS/kP8AGtaiseV9zo9pT/l/H/gGT/bNv/cl/If40f2zb/3JfyH+Na1FHK+4e0p/y/j/AMAyf7Zt/wC5L+Q/xo/tm3/uS/kP8a1qKOV9w9pT/l/H/gGT/bNv/cl/If40f2zb/wByX8h/jWtRRyvuHtKf8v4/8Ayf7Zt/7kv5D/Gj+2bf+5L+Q/xrWoo5X3D2lP8Al/H/AIBk/wBs2/8Acl/If41xXj3WreDWPD986SmKznaaQADcVVoycc9eK9LrivGn/I5eDv8Ar/H/AKMirfDxftN+j/JnPiKlP2fw9V1815FH/hdXh3/ny1T/AL9R/wDxdH/C6vDv/Plqn/fqP/4uvXKK8X6vV/n/AA/4J9L9fy3/AKBn/wCDP/tTyP8A4XV4d/58tU/79R//ABdH/C6vDv8Az5ap/wB+o/8A4uvXKxm8YeGlYq3iLSQwOCDfR8f+PVLo1FvUX3f8E0p4rA1Pgwkn6Tb/APbDzz/hdXh3/ny1T/v1H/8AF0f8Lq8O/wDPlqn/AH6j/wDi69UuNQsrSzF3dXkEFsQCJpJVVCDyPmJxTbDVNP1SJpNMvra8jU4Z7eZZAD9QTT9jUvb2i+7/AIIvreA5eb6pK3fnf/yB5b/wurw7/wA+Wqf9+o//AIuj/hdXh3/ny1T/AL9R/wDxdeuUU/q9X+f8P+CR9fy3/oGf/gz/AO1PI/8AhdXh3/ny1T/v1H/8XR/wurw7/wA+Wqf9+o//AIuvXKKPq9X+f8P+CH1/Lf8AoGf/AIM/+1PI/wDhdXh3/ny1T/v1H/8AF0f8Lq8O/wDPlqn/AH6j/wDi69coo+r1f5/w/wCCH1/Lf+gZ/wDgz/7U8j/4XV4d/wCfLVP+/Uf/AMXR/wALq8O/8+Wqf9+o/wD4uvXKKPq9X+f8P+CH1/Lf+gZ/+DP/ALU8cvfi54T1G0e1v9L1C4gkGGjkgjIP/j9cRH4wt/DGr/bPAVzqFtBI2ZbG+RWiP5Oc/wAx619NUVEsJUk78+vp/wAE7MNnuDw0XCGHbi9053T+Tj+Jwng34r6N4n8u1vCNN1FuPJlb5JD/ALDf0PP1ru6KK7KanGNpu/4Hz2LqYerUcsPTcF2vzfc7L9QooorQ5AooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAK5zxR4I0rxRGXuE+z3qj5LuIYYemf7w+v4EV0dFXCcoPmi7MmUIzXLJXR5UNX8QeBbhbTxRC9/ppO2K/i+Yge57/Q8+hNdnp+o2mqWa3WnzpPC3RkPT2I7H2Nb09vDdW7wXMSTRSDDxuoZWHoQa881bwBqGhXb6r4EuGibrJYSNlXHoM9fofwNdkalOt8Xuy/B/5HK41KO3vR/Ff5nXUVy2geOLXUrj7BqsR03U1O1oJcgM3sT0PsefrXU1E4Sg7SRpCcZq8WFFFFQWFFFFABRRRQAVxXjT/kcvB3/X+P/RkVdrXFeNP+Ry8Hf9f4/wDRkVb4f+J8n+TMMR/D+780enUUUV552BXhXhW80Twtpd0njjwhdyNNfO8d7c6UrxqhCgLvfnqGOAD1r3OTf5TeVt8zB27ume2a848SJ498T6DNoE3huytBcFVmvxfq0eAwOVT7w6d81x4mL0kt1e2lz6LJaqSnSm0oSceZ86g0lfVd99lc66fS/DviPT7G+ubW0vbKCPzLZpBmJUIHO08YwB1HGK82+GdtFcfFfXr/AMPR+VocavGCnEbEsNoA9OGI9BWv4u8M+KLfwJpXhbwrAbu3WLZfTrMkTOBj5RuI4JJ9emPrreCRrekRWeit4K/snTkDeZdnU4pm3YJ3MqgFiSAPbPoKzl79WN42t1s9+1+x2UpLD5fW9nVU+e6UXOK5Yp3vytp8z6JLz62O3ooor0D5EKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAwPE3g3SvFNvi+i8u5UYjuoxh1/xHsf0riGvfEfgCVYNdjfVNHztjvI+WQdgc/yP4GvVqbLFHNE0UyLJG4wyOMhh6EV008Q4rllrHt/kc9SgpPmi7P8Arc5zTNVstYs1udOuEniPXaeVPoR1B+tXK5TWfh3daZeNqvga4NpcdXs2b5JPYZ4/A8e4puh+OYbq6/s3X4Tp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data-image-state="image-loaded"></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">-----------------------</div><div style="block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">NOTE: <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "> There can be more than one PPT with smallest area for the same value of <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">R<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">. <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">For example if <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">R = 4.5<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">, there are two possible smallest area PPT: <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">[20 21 29] <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">and <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">[12 35 37]<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">. Both have an area of <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">210<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">. <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">In cases where there are more than one possible PPT's, the function should output the PPT with the least perimeter.<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "> So, for <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">R = 4.5<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">, the function output should be: <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">[20 21 29]<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; ">.</div></div></div>

Pythagorean Triangle is a right triangle all sides of which are represented by integers. Examples are triangles with sides  and . 
We say that pythagorean triangle is a primitive pythagorean triangle (PPT), if all of its sides are relatively prime. That is, if the sides of a pythagorean triangle are a, b and c, it is PPT if and only if: gcd(a,b) == 1 && gcd(a,c) == 1 && gcd(b,c) == 1.
Given a circle of radius R, find the dimensions of the PPT with the smallest area, that can enclose the circle, without touching the circle (i.e. no sides is a tangent of the circle, or crosses the circle). For example, in the figure below, if R = 12, the smallest pythagorean triangle that can enclose the circle without touching it is [39,52,65]. However this is not primitive since, gcd(39,52) = 13 ≠ 1. The correct smallest PPT is the triangle with dimensions: [48 55 73].
                                                   
-----------------------
NOTE:   There can be more than one PPT with smallest area for the same value of R. For example if R = 4.5, there are two possible smallest area PPT: [20 21 29] and [12 35 37]. Both have an area of 210. In cases where there are more than one possible PPT's, the function should output the PPT with the least perimeter. So, for R = 4.5, the function output should be: [20 21 29].

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Problem 57954. Easy Sequences 108: Enclosing a Circle with Primitive Pythagorean Triangles

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