Solving geometric problem

Question

I want to find the coordinates of the $p$ point and $\beta$ angle in the following figure. The point is defined by the angle $\alpha$, the positions of the $a$ point, and the radius of the circle $r$, so the known variables are:

• $a$ - Coordinates
• $r$ - Radius
• $\alpha$ angle

And the unknown variables

• $p$ coordinates
• $\beta$ angle

Answer 1

Let $O(0,0), A(a,0)$ where $a\lt 0$.

Now consider $\triangle{OAP}$. Note that $\angle{PAO}=\pi-\alpha,\angle{OPA}=\pi-\beta-(\pi-\alpha)=\alpha-\beta$.

By the law of sines, $$\frac{r}{\sin(\pi-\alpha)}=\frac{-a}{\sin(\alpha-\beta)}$$ Hence, we have $$\sin(\alpha-\beta)=\frac{-a}{r}\sin(\pi-\alpha)$$ So, $$\alpha-\beta=\arcsin\left(\frac{-a}{r}\sin(\pi-\alpha)\right),$$ i.e. $$\beta=\alpha-\arcsin\left(\frac{-a}{r}\sin(\pi-\alpha)\right)$$

Then, finally, note that we can write $P(r\cos(\pi-\beta),r\sin(\pi-\beta))$.