Is there a way for me to proof the mid-point theorem without using the triangle congruence theorems?
I can't seem to find one that's not using the triangle congruence theorems. The reason I'm trying to find the mid-point theorem without using the triangle congruence theorems is because, I'm reading about the proof for the triangle congruence theorem (specifically SAS) and they use the mid-point theorem to prove it.
Use vectors.
For triangle ABC let $\vec {AB}=p$ and let $\vec {AC}=b$.
Let midpoint of AB be M and let midpoint of AC be N.
Then $\vec {AM}=\frac p 2$ and $\vec {AN}=\frac q 2$
$\vec {BC}=\vec {AC}-\vec{AB}=q-p$
$\vec {MN}=\vec {AN}-\vec{AM}=\frac q 2-\frac p 2=\frac 1 2 (q-p)=\frac 1 2 \vec {BC}$
Thus MN is parallel to BC and half the magnitude.