Angle of intersection lines

Question

I have got $3$ points say $(x_1,y_1)$, $(x_2,y_2)$ and the point of intersection $(x,y)$. I need to know the angle of convex angle of intersection of points. I need to whether the angle of intersection is acute or obtuse. The lines can be of any orientation. I need this to find which angle bisector I should choose.

Answer 1

Hint: It is the angle between the vectors $$u=(x-x_1,y-y_1),\ \ v=(x-x_2,y-y_2)$$ It is $$\cos\theta=\frac{u\cdot v}{||u||.||v||}$$

Added

The slope of the bisectors of the angle between two lines with slope $m_1,m_2$ is $$m=\frac{a}{b+\sqrt{a^2+b^2}}\ \ \text{and}\ \ m=\frac{a}{b-\sqrt{a^2+b^2}}$$ where $a=m_1+m_2$ and $b=1-m_1m_2$

Answer 2

Use the fact that locus of distance from two lines is same is angle bisector.

Acute angle bisector : if $2\theta<\frac{\pi}2$ then $\tan\theta<1$

$\theta$ is angle between bisector and one of the lines.

Angle between 2 lines :

How to find the angle between two straight line equations?