{"id":626,"date":"2013-04-20T20:26:33","date_gmt":"2013-04-21T01:26:33","guid":{"rendered":"http:\/\/daylateanddollarshort.com\/bloog\/?p=626"},"modified":"2013-04-20T20:34:12","modified_gmt":"2013-04-21T01:34:12","slug":"a-ceva-like-theorem-for-tetrahedra","status":"publish","type":"post","link":"http:\/\/daylateanddollarshort.com\/bloog\/a-ceva-like-theorem-for-tetrahedra\/","title":{"rendered":"A Ceva-like Theorem for Tetrahedra"},"content":{"rendered":"<p>A question at <a href=\"http:\/\/math.stackexchange.com\/q\/366456\/409\" title=\"Math.StackExchange\">Math.StackExchange<\/a> asked about <a href=\"http:\/\/math.stackexchange.com\/q\/366456\/409\" title=\"Ceva's Theorem in three dimensions\">&#8220;Ceva&#8217;s Theorem in three dimensions&#8221;<\/a>.<\/p>\n<p>So <a href=\"http:\/\/math.stackexchange.com\/a\/366846\/409\" title=\"My 'Ceva Theorem in three dimensions' answer on Math.StackExchange\">I derived one.<\/a><\/p>\n<p>The result (which I believe may be new, although it could very well exist in the hundreds of years of mathematical literature since Ceva) replaces the traditional ratios of segment lengths with &#8220;triple-ratios&#8221; of triangle areas, to satisfying effect. Whereas the ratios in Ceva&#8217;s Theorem reduce to \\(1\\) (or &#8220;\\([1:1]\\)&#8221;), my triple-ratios reduce to \\([1:1:1]\\).<\/p>\n<p>I like the result enough that I&#8217;ve tidied the derivation, added a smidgeon of context, and touched-on the (current) lack of a corresponding Menelaus-like theorem. See my note: <a href=\"http:\/\/dlnds.com\/mathdocs\/A-Ceva-Like-Theorem-for-Tetrahedra.pdf\" title=\"'A Ceva-like Theorem for Tetrahedra' at the Bloog!\">&#8220;A Ceva-like Theorem for Tetrahedra&#8221; (PDF)<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A question at Math.StackExchange asked about &#8220;Ceva&#8217;s Theorem in three dimensions&#8221;. So I derived one. The result (which I believe may be new, although it could very well exist in the hundreds of years of mathematical literature since Ceva) replaces the traditional ratios of segment lengths with &#8220;triple-ratios&#8221; of triangle areas, to satisfying effect. Whereas [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7,6,11],"tags":[],"class_list":["post-626","post","type-post","status-publish","format-standard","hentry","category-classroom","category-misc-math","category-open-question"],"_links":{"self":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/626","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/comments?post=626"}],"version-history":[{"count":10,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/626\/revisions"}],"predecessor-version":[{"id":639,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/626\/revisions\/639"}],"wp:attachment":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/media?parent=626"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/categories?post=626"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/tags?post=626"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}