Mid-segment Theorem

Date: 02/02/99 at 19:55:07
From: Richard Phung
Subject: Mid-segment theorem

The Mid-segment theorem states that if a segment bisects two sides of 
a triangle then the segment is parallel and is half of the opposite 
side. Can you help me prove it?

Thank you.
Richard Phung

Date: 02/03/99 at 02:37:03
From: Doctor Floor
Subject: Re: Mid-segment theorem

Hi Richard,

I will give you a few hints that should make it possible to prove the 
theorem. You can go over writing a two-column proof yourself after 
that.

Think of a triangle ABC, with D is the midpoint of AC and E is the 
midpoint of BC.

          C
         / \
        D---E
       /     \
      A-------B

Now draw a triangle D'E'C' that is twice as big as DEC beside ABC.

          C           C'
         / \         / \
        D---E       /   \
       /     \     /     \
      A-------B   D'------E'

Since D'C'= AC (length) and C'E' is BC (length) and angle C is angle C' 
we have that triangles ABC and D'E'C' are congruent (SAS).

Since triangle DEC is half triangle D'E'C' we know that DE must be half 
the length of AB.

Since D'E'C' is congruent to ABC, DEC is similar to ABC. So angle(CDE) 
is equal to angle A and DE must be parallel to AB.

I hope this helps!

Best regards,

- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/