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November 13th, 2010, 12:27 PM  #1 
Senior Member Joined: Apr 2009 Posts: 106 Thanks: 0  Similar Triangles
Use one of the conditions for similarity of triangles (AA, SSS, or SAS) to prove that a midsegment of a triangle is parallel to a side of the triangle and is half the length of that side.

November 14th, 2010, 07:40 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,725 Thanks: 1536 
The midsegment joins the midpoints of two sides, thus "cutting off" a triangle that shares an angle with the original triangle and has sides enclosing that angle that are half as long as the corresponding sides of the original triangle, so the triangles are similar (SAS). Can you finish from there? The theorem is known as the triangle midsegment theorem (or, in older books, the midpoint theorem). There's a generalization where the proportionality constant is any value between 0 and 1. 

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