{"id":23,"date":"2013-09-07T18:01:24","date_gmt":"2013-09-08T01:01:24","guid":{"rendered":"http:\/\/celloexpressions.com\/ts\/?page_id=23"},"modified":"2020-06-09T19:29:14","modified_gmt":"2020-06-10T02:29:14","slug":"proof","status":"publish","type":"page","link":"https:\/\/celloexpressions.com\/ts\/dynamic-documentation\/proof\/","title":{"rendered":"3. A Geometric Proof"},"content":{"rendered":"<h3>Proof: Angle HEG has measure theta<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright\" src=\"\/ts\/wp-content\/uploads\/sites\/3\/2013\/09\/proof-init.jpg\" alt=\"\" width=\"417\" height=\"317\"><\/p>\n<h5>Figure 1\u2014The Base Figure:<\/h5>\n<p>Here we have two circles whose center points are A and C and whose intersection points are E and F. Points G and H are on circles C and A respectively. We have constrained line HG to go through point F and angles AEC and CEG are constrained as theta and beta. Our objective is to prove that angle HEG is always theta also when the above conditions are met.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<h3><em>Geometry Expressions<\/em>&nbsp;Proof:<\/h3>\n<p>Draw the figure with all of the above constraints. Calculate angle HEG. If the output is theta, then (assuming&nbsp;Geometry Expressions&nbsp;is correct) we have proven that the angle is theta.&nbsp;This video goes through the proof with Geometry Expressions:<\/p>\n<div style=\"width: 640px;\" class=\"wp-video\"><video class=\"wp-video-shortcode\" id=\"video-23-1\" width=\"640\" height=\"360\" preload=\"metadata\" controls=\"controls\"><source type=\"video\/mp4\" src=\"http:\/\/celloexpressions.com\/ts\/wp-content\/uploads\/sites\/3\/2013\/09\/gx-proof.mp4?_=1\" \/><source type=\"video\/webm\" src=\"https:\/\/celloexpressions.com\/ts\/wp-content\/uploads\/sites\/3\/2013\/09\/gx-proof.webm?_=1\" \/><a href=\"http:\/\/celloexpressions.com\/ts\/wp-content\/uploads\/sites\/3\/2013\/09\/gx-proof.mp4\">http:\/\/celloexpressions.com\/ts\/wp-content\/uploads\/sites\/3\/2013\/09\/gx-proof.mp4<\/a><\/video><\/div>\n<h3>Interactive Version<\/h3>\n<p><iframe loading=\"lazy\" style=\"border: 2px solid #004; margin: 0 auto;\" src=\"https:\/\/euclidsmuse.com\/e?id=97&amp;height=734\" width=\"80%\" height=\"734\"><\/iframe><\/p>\n<h3>Formal\/traditional Proof:<\/h3>\n<p>To prove that angle HEG is theta, first prove that angle AEH is beta. To do this, we show that triangles HAE and GCE are isosceles because two of their sides are radii and the third is a chord. Therefore, angle CGE is also beta and angle ECG is pi-(2*beta) because of the triangle angle sum theorem. Then, the <a href=\"https:\/\/euclidsmuse.com\/members\/cello\/apps\/app\/263\" target=\"_blank\" rel=\"noopener noreferrer\">chord angle theorem<\/a> tells us the value of angle EFG. In this case, angle EFG is (pi\/2)-beta. Additionally, because GFH is a line, angle EFH is (pi\/2)+beta.<\/p>\n<h5>Figure 2: circle C angles<\/h5>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"\/ts\/wp-content\/uploads\/sites\/3\/2013\/09\/proof-circle-c.jpg\" alt=\"circle c angles\" width=\"309\" height=\"295\"><\/p>\n<p>Using the <a href=\"https:\/\/euclidsmuse.com\/members\/cello\/apps\/app\/262\" target=\"_blank\" rel=\"noopener noreferrer\">cyclic quadrilateral theorem<\/a>, we know angle HIE from angle HFE. Then we again use the chord angle theorem to find angle HAE. Because the triangle is isosceles, we have proven that angle AEH is beta (figure 3). Finally, we know through the addition of angles AEC and CEG that angle AEG is equal to beta+theta and that angle AEH is beta, so angle HEG must be theta to satisfy that angle AEG is beta+theta (figure 4).<\/p>\n<h5 style=\"text-align: left;\">Figure 3\u2014circle A angles:<\/h5>\n<h5 style=\"text-align: right;\">Figure 4\u2014final figure:<\/h5>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft\" src=\"\/ts\/wp-content\/uploads\/sites\/3\/2013\/09\/proof-circle-a.jpg\" alt=\"\" width=\"293\" height=\"303\" border=\"0\"><img loading=\"lazy\" decoding=\"async\" class=\"alignright\" src=\"\/ts\/wp-content\/uploads\/sites\/3\/2013\/09\/proof-final.jpg\" alt=\"\" width=\"437\" height=\"340\" border=\"0\"><\/p>\n<table width=\"100%\">\n<tbody>\n<tr>\n<td style=\"text-align: left;\"><a title=\"2. The Top View\" href=\"http:\/\/celloexpressions.com\/ts\/dynamic-documentation\/top-view\/\">Previous<\/a><\/td>\n<td style=\"text-align: right;\"><a title=\"4. The Side View \u201c3d\u201d Model\" href=\"http:\/\/celloexpressions.com\/ts\/dynamic-documentation\/side-view\/\">Next<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>Proof: Angle HEG has measure theta Figure 1\u2014The Base Figure: Here we have two circles whose center points are A and C and whose intersection points are E and F. Points G and H are on circles C and A respectively. We have constrained line HG to go through point F and angles AEC and [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":15,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"page-main-content.php","meta":{"footnotes":""},"class_list":["post-23","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/celloexpressions.com\/ts\/wp-json\/wp\/v2\/pages\/23","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/celloexpressions.com\/ts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/celloexpressions.com\/ts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/celloexpressions.com\/ts\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/celloexpressions.com\/ts\/wp-json\/wp\/v2\/comments?post=23"}],"version-history":[{"count":4,"href":"https:\/\/celloexpressions.com\/ts\/wp-json\/wp\/v2\/pages\/23\/revisions"}],"predecessor-version":[{"id":198,"href":"https:\/\/celloexpressions.com\/ts\/wp-json\/wp\/v2\/pages\/23\/revisions\/198"}],"up":[{"embeddable":true,"href":"https:\/\/celloexpressions.com\/ts\/wp-json\/wp\/v2\/pages\/15"}],"wp:attachment":[{"href":"https:\/\/celloexpressions.com\/ts\/wp-json\/wp\/v2\/media?parent=23"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}