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PS

Here is a way of showing that our first conjecture (Method A) is false,
ie it is NOT the case that DA bisects the angle at A when D is the midpoint of BC.

Here F is a point such that AF = AB (ie ABF is isosceles).
So, if DA does bisect the angle at A, then E will be a midpoint of BF.
But if E and D are both midpoints, then ED must be parallel to FC,
which means ED produced will not go through A...

You might be able to find a better way of formulating an argument based on this situation ...

The end