Proving Midpoint of Triangle Sides is Parallel and 1/2 Length

AI Thread Summary
To prove that the line segment joining the midpoints of two sides of a triangle is parallel to and half the length of the third side, one can utilize the concept of similar triangles. The Basic Proportionality Theorem establishes that if two lines are parallel, they divide the other two sides proportionally. In the context of triangle ABC, if line segment DE is drawn parallel to side BC, it divides sides AC and AB in the same ratio as BC to DE. This leads to the conclusion that the segment connecting the midpoints is indeed parallel and half the length of the third side, confirming the Midpoint Theorem as a direct consequence of the Basic Proportionality Theorem. Understanding these relationships is essential for geometric proofs involving triangles.
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how do you prove that the line segment joing the midpoint of two sides of a triangle is parallel to and 1/2 the length of the 3rd side?
 
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Try using "similar triangles".
 
I think Basic proportionality theorem is proved before midpoint theorem. Basic proportionality theorem states that if in TrABC, BC||DE, then D and E divides AC and AB in the ratio, same as that of BC:DE.
 
The midpoint theorem is a direct conxequence of Baisproportionality theorem.
 

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