Bisector equation

Question

I have the coordinates of the triangle vertices (A(x1,y1),B(x2,y2),C(x3,y3)) . Could I write the bisector equation taken from the top of A (AM, M(x,y))? Thanks in advance.

Answer 1

Suppose $M(x,y)$ is on the bisector of angle $\angle BAC$, we have $$\cos\angle BAM=\frac{AM\cdot AB}{||AM||\cdot ||AB||}=\frac{AM\cdot AC}{||AM||\cdot ||AC||}=\cos\angle CAM$$ Or, $$AM=t\left(\frac{AB}{||AB||}+\frac{AC}{||AC||}\right)$$