{"id":95,"date":"2017-05-07T11:18:36","date_gmt":"2017-05-07T11:18:36","guid":{"rendered":"https:\/\/petermc.net\/blog\/?p=95"},"modified":"2017-05-07T11:22:21","modified_gmt":"2017-05-07T11:22:21","slug":"the-medians-of-a-triangle-are-concurrent","status":"publish","type":"post","link":"https:\/\/petermc.net\/blog\/2017\/05\/07\/the-medians-of-a-triangle-are-concurrent\/","title":{"rendered":"The medians of a triangle are concurrent"},"content":{"rendered":"

In case anyone reading this does not know, a median is a line connecting a vertex of a triangle to the midpoint of the opposite edge. The theorem is that the three medians of a triangle are concurrent (i.e. they meet in a single point). Here are four proofs.<\/p>\n

Proof 1:<\/strong> (Transformation geometry)
\n\"\"
\nLet E and F be the midpoints as shown and let BE and CF intersect at G. Consider the dilation about A with factor 2. It sends E to C, F to B and G to Q (this is the definition of Q). Then EG and CQ are parallel, as are FG and BQ. Thus BGCQ is a parallellogram and the diagonals of a parallelogram bisect each other QED.<\/p>\n

Proof 2:<\/strong> (Vectors. Efficient and boring). Write \"{\bf, \"{\bf and \"{\bf for A, B and C respectively. Let G be \"({\bf. It is easy then to check that the midpoint D of AB is \"({\bf and that A, D and G are concurrent.<\/p>\n

Proof 3:<\/strong> (Why not just prove Ceva’s Theorem)\"\"<\/p>\n

Ceva’s Theorem states that in the situation shown, AD, BE and CF are concurrent if and only if<\/p>\n

\"\displaystyle <\/p>\n

To prove this, note that in the case of concurrence<\/p>\n

\"\displaystyle <\/p>\n

The rest of the proof is routine.<\/p>\n

Proof 4:<\/strong> (my favourite) WLOG the triangle is equilateral. Now the statement is obvious (e.g. by symmetry).<\/p>\n

Perhaps some elaboration should be made to the WLOG. An affine transformation of \"\mathbb{R}^2\" is a map of the form \"{\bf{v}}\mapsto where \"A\" is an invertible matrix and \"{\bf{b}}\" is a vector. The affine transformations act transitively on the set of (nondegenerate) triangles and the property of having concurrent medians is clearly invariant under these transformations.<\/p>\n

Acknowledgements:<\/strong> Thanks to Inna Lukyanenko<\/a> for the first proof and tikz files, and to pdftoppm<\/code> for converting the .pdf output to .png.<\/p>\n","protected":false},"excerpt":{"rendered":"

In case anyone reading this does not know, a median is a line connecting a vertex of a triangle to the midpoint of the opposite edge. The theorem is that the three medians of a triangle are concurrent (i.e. they … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6],"tags":[],"class_list":["post-95","post","type-post","status-publish","format-standard","hentry","category-maths"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":false,"jetpack_shortlink":"https:\/\/wp.me\/p7V6a7-1x","_links":{"self":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/95","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/comments?post=95"}],"version-history":[{"count":10,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/95\/revisions"}],"predecessor-version":[{"id":111,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/95\/revisions\/111"}],"wp:attachment":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/media?parent=95"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/categories?post=95"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/tags?post=95"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}