A non euclidean line in $\mathbb{RP}^1$ in terms of reflections about the unit circle can be written in the form
$A+B(\overline{w}+w)+C(\overline{w}w)=0$
Where $w=\frac{1}{\overline{z}}$
The equation I'm working with is $w+\overline{w}-2w\overline{w}=0$ and I am asked to verify that this is the equation of a semicircle with endpoints $O$ (the center of the unit cirlce) and 1 on the x axis. I was thinking that the coefficients would get me somewhere but I'm not too sure.
If we write $w=x+iy$ with $x,y\in\mathbb R$ then $\bar w+w=2x$ and $\bar ww=x^2+y^2$. So you have the equation $A+2Bx+Cx^2+Cy^2=0$. In your specific case, $A=0,B=1,C=-2$ you have
\begin{align*} 2x - 2x^2 - 2y^2 &= 0 \\ x^2+y^2-x &= 0 \end{align*}
Completing the square you have
\begin{align*} x^2-2x\tfrac12+\left(\tfrac12\right)^2+y^2&=\left(\tfrac12\right)^2 \\ \left(x-\tfrac12\right)^2+y^2&=\left(\tfrac12\right)^2 \end{align*}
which is the equation of a circle with center at $\left(\tfrac12,0\right)$ and radius $\tfrac12$. If you restrict it to the upper half plane, you get a semicircle of course.