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I giving a second try to this question. Hopefully, with better problem definition.

I have a circle inscribed inside a square and would like to know the point the radius touches when extended. In the figure A, we have calculated the angle(θ), C(center) , D and E. How do i calculate the (x,y) of A and B?

enter image description here

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In the case you've drawn, you already know the $x$ value, assuming the circle has center in $(C_x,C_y)$ and radius $r$, $A_x=B_x=C_x+r.$ As for the $y,$ a little trigonometry helps: $A_y=C_y+r·\tan \theta.$

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If you know the coordinates of the center then you add $r$ to the $x$ coordinate and you add $r \tan (\theta)$ to the $y$ coordinate of the center to get coordinates of $A$

Similarly you can find coordinates of $B$

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Describe the circle as $$\vec x=\vec m+\begin{pmatrix} r\cos(t)\\ r\sin(t) \end{pmatrix}.$$ Now consider the ray $$\vec y=\vec m+\lambda\begin{pmatrix} r\cos(t)\\ r\sin(t) \end{pmatrix}$$ with $\lambda>0$. You want to have the first coordinate for $t\in(-\pi/2,\pi/2)$ of $\vec y$ to be $m_1+r$, hence $\lambda=1/\cos(t)$ and the desired point is $$\vec m+\frac{1}{\cos(t)}\begin{pmatrix} r\cos(t)\\ r\sin(t) \end{pmatrix}=\begin{pmatrix} m_1+r\\ m_2+r\tan(t) \end{pmatrix}.$$

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