One approach to drawing the bisector of $\angle BAC$ with tikz is to define two coordinates, say $AtB$ and $AtC$ on $AB$ and $AC$, respectively, at equal distance from $A$. Then one computes the midpoint of the segment connecting $AtB$ to $AtC$; finally, one draws the line from $A$ to that midpoint. The code may look like this:
\coordinate (AtB) at ($ (A) !1cm! (B) $);
\coordinate (AtC) at ($ (A) !1cm! (C) $);
\coordinate (BACb) at ($ (AtB) !0.5! (AtC) $);
\draw (A) -- (BACb);