# Equation of perpendicular line from the midpoint of a chord to a tangent on a unit circle (complex numbers)

As the title suggests, I would like the find the equation of the perpendicular line from $$M_1$$ of the side $$A_2A_3$$ to the tangent line at $$A_1$$. Without a loss of generality, suppose that on the unit circle, the coordinates of $$A_1$$, $$A_2$$ and $$A_3$$ are given by $$u_1$$, $$u_2$$ and $$u_3$$ respectively.

It is well known that the complex form of a line passing through 2 distinct point $$p_1$$ and $$p_2$$ is given by $$z+ p_1p_2\bar{z}=p_1+p_2.$$
Since the tangent to the unit circle at $$A_1$$ coincides, it is clear that the equation of tangent at $$A_1$$ is given by $$z+u_1^2\bar{z}=2u_1.$$ How can I find the equation of the perpendicular line from $$M_1$$(where $$M_1=\dfrac{u_2+u_3}{2})$$ of the side $$A_2A_3$$ to the tangent line at $$A_1$$?

When I was reading the proof, the author gave the required perpendicular line to be $$z-u_1^2\bar{z}=\frac{1}{2}[(u_2+u_3)-u_1^2(\bar{u_2}+\bar{u_3})].$$

Can someone explain to me how is this obtain? I am mainly confused as to how the sign between $$z$$ and $$\bar{z}$$ changed from positive to negative. Also, the author seem to ignore the "constant" terms on the RHS (That is, the "$$p_1+p_2$$" constant for line equations) and simply subbed in $$M_1=\dfrac{u_2+u_3}{2}$$ into the LHS. Any help would be appreciated!

The equation of a perpendicular line from $$C(c)$$ to a line $$AB$$ where $$A(a),B(b)$$ with $$|a|=|b|=1$$ and $$a\not =b$$ is given by \begin{align}\frac{z-c}{a-b}\in\mathbb Ri &\iff (z-c)(\bar a-\bar b)\in\mathbb Ri \\\\&\iff (z-c)\left(\frac 1a-\frac 1b\right)+(\bar z-\bar c)(a-b)=0 \\\\&\iff z-ab\bar z=c-ab\bar c\end{align}
Now, set $$a=u_1,c=\frac{u_2+u_3}{2}$$ and take $$b\to a$$.