What is the relationship between the line segment joining the midpoint at two sides and the third side of a triangle?

This is not quite right. Firstly, as an aside, we don't talk about "lengths of points". As such, your first statement about the length of $A$ doesn't quite make sense.

You then make a statement about the line $DE$. You say that it's "half-way between" something and something else. What is the meaning of this? What does it mean for a whole line to be halfway between something and something else? If $DE$ was not a straight line, would this statement still stand? What if $DE$ was "rotated" a little?

Certainly your final assertion isn't right. What is the logical step between $DE$ being halfway between $A$ and $BC$, and $DE$ having half the length of $BC$? This does not follow in any simple way.

As such, your proof has a few holes, and I suspect is not quite going to be fruitful enough to prove your statement. I hope this helps.

Is parallel to the third side but it's dimension is not determinable

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Is half as long as the third side but can't say if it is parallel to it

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Is parallel to and half as long as the third side

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