{"id":116,"date":"2022-05-22T10:44:29","date_gmt":"2022-05-22T10:44:29","guid":{"rendered":"https:\/\/convexabacus.xyz\/?p=116"},"modified":"2022-05-22T11:33:21","modified_gmt":"2022-05-22T11:33:21","slug":"two-discs-in-an-envelope","status":"publish","type":"post","link":"https:\/\/convexabacus.xyz\/index.php\/2022\/05\/22\/two-discs-in-an-envelope\/","title":{"rendered":"Two Discs in an Envelope"},"content":{"rendered":"\n<p>Two circles are inscribed in a rectangle as shown. One diagonal passes through the point of tangency where the two circles touch, and the other diagonal is tangent to the smaller circle.<\/p>\n\n\n\n<p>You may define the radius of the large circle as 1 and the radius of the small circle as <em>r<\/em>. Can you find an equation for <em>r<\/em>, and numeric values for <em>r<\/em> and for the lengths of the sides of the rectangle?<\/p>\n\n\n\n<p>I created this construction using Geogebra, iteratively adjusting it until everything fitted. The aspect ratio of the rectangle turns out to be about 1.72, a fact that can be used to check your own result. The dimensions of the rectangle are not an initial condition, but a consequence of the radii of the two circles. However, as an alternative exercise try drawing this construction yourself, beginning with a 100 by 172 rectangle.<\/p>\n\n\n\n<p>I am delighted to report that this problem was solved independently by Ted Courant and by Keith Raskin after I posted it on the Math, Math Education, Math Culture group on LinkedIn. They also observed that the rectangle is more accurately drawn with side lengths 1003 and 1725.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"568\" src=\"https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/05\/CirclesFitEnvelope-1024x568.png\" alt=\"\" class=\"wp-image-117\" srcset=\"https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/05\/CirclesFitEnvelope-1024x568.png 1024w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/05\/CirclesFitEnvelope-300x167.png 300w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/05\/CirclesFitEnvelope-768x426.png 768w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/05\/CirclesFitEnvelope-1536x852.png 1536w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/05\/CirclesFitEnvelope-2048x1137.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Two circles are inscribed in a rectangle as shown. One diagonal passes through the point of tangency where the two circles touch, and the other diagonal&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8,5,11],"tags":[],"class_list":["post-116","post","type-post","status-publish","format-standard","hentry","category-geogebra","category-geometry","category-mathematics"],"_links":{"self":[{"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts\/116","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/comments?post=116"}],"version-history":[{"count":3,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts\/116\/revisions"}],"predecessor-version":[{"id":120,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts\/116\/revisions\/120"}],"wp:attachment":[{"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/media?parent=116"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/categories?post=116"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/tags?post=116"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}