Proving that a line segment joining the midpoints of two sides of a triangle equals half of third side.

Question

Is this proof correct? I know it can be proven using the Midpoint Theorem. I believe the proof is correct, in that case, please explain how I should prove it with the same approach without a graph. If it is incorrect, please explain why.

What I am basically trying to do involves interpreting:

(source)

on a graph such that CA is on the x-axis.
As point D lies in the centre of slope BA, y position for D should equal
(y position for B + y position for A)/2
=(y position for B)/2

y position for D = ED
y position for B = BC

Therefore, ED = 1/2 BC
Would appreciate if you could stick to layman terms.

EDIT1: Changed how I was expressing my doubt.

Answer 1

Alternative proof:

Since $\angle BAC = \angle DAE$ and ${BA\over DA} = {CA\over EA} =2$ triangles $ABC$ and $ADE$ are similar with ratio of similarity 2, so $BC = 2 DE$.

Answer 2

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.

Answer 3

"Since the length of A is 0 (A point has no dimensions)"

Many people would object to this. Strictly speaking, length of a point is not what you are thinking it like to be.

"DE is halfway between A and BC "

Again, this is not what a mathematician would accept. What do you mean by the midpoint of a point (here $A$) and a line (here $BC$). It could mean the point that it halfway between the line passing through $A$ and intersecting $BC$ perpendicularly. But,it may not mean this also. There ambiguity. So its not acceptable.

"DE is halfway between A and BC "

You should be careful using mathematical logic. The therefore has great power of conclusion to a situation. And with great power comes great responsibility. Your previous statement and the therefore statement is not complying. Its difficult to see why you make such a conclusion. So that's again not acceptable.