What is the slope of a line half way between two lines of slope m1 and m2? I have two lines of slope m1 and m2 respectively.  What is the slope of the line bisecting these two lines?  Clearly there are 2 different lines which can be constructed, one perpendicular to the other.  I am looking for the one which bisects the acute angle between the 2 lines.

We have:
$$m_1=\frac{y_2-y_1}{x_2-x_1}=\tan{(\alpha)}$$
And:
$$m_2=\frac{y_3-y_1}{x_3-x_1}=\tan{(\beta)}$$
Therefore,
$$m_3=\tan\left(\beta+\frac{\alpha-\beta}{2}\right)=\tan\left(\frac{\alpha+\beta}{2}\right)$$
Where $m_3$ is the slope of the bisector.
Now, we will use this result to find an expression in terms of $m_1$ and $m_2$.
One may show that:
$$\tan\left(\frac{\alpha+\beta}{2}\right)\equiv \frac{1-\cos(\alpha+\beta)}{\sin(\alpha+\beta)}$$
Now, you can use the double angle formulas for $\sin(\alpha+\beta)$ and $\cos(\alpha+\beta)$, then apply the substitutions $\sin(x)=\frac{m}{\sqrt{1+m^2}}$ and $\cos(x)=\frac{1}{\sqrt{1+m^2}}$ to obtain an expression for $m_3=f(m_1,m_2)$.
