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The Midline Theorem is a geometry theorem that states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half of its length. This theorem can also be stated as the midline of a triangle being half the length of the third side.
The Midline Theorem can be proven using basic geometry principles. Here is a simple proof:
Let's consider a triangle ABC with midpoints D and E on sides AB and AC, respectively.
Draw segment DE to connect the midpoints of AB and AC.
Connect points D and E to vertex B and C, respectively.
Notice that triangles ADE and ABC are similar, as they share the same angles.
Using the similarity of these triangles, we can set up the following proportion:
AD / AB = AE / AC
Rearranging this proportion, we get:
AB / AD = AC / AE
By the Transitive Property of Equality, we can write:
AB / AD = AC / AE = BC / DE
This shows that segment DE is parallel to BC, since AB / AD = BC / DE, and that DE is half the length of BC, since AB + AC = 2(DE).
Therefore, we have proved that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half of its length, which is known as the Midline Theorem.