how to find circle circumference point from a point inside the circle

Question

given the image bellow how can i find the points of a circles circumference from a point inside the circle with given X and Y ? also can how to calculate it from a point outside the cirlce ?

https://i.stack.imgur.com/0fv9G.png

Answer 1

If $(x_0, y_0)$ is the centre of the circle and if $R$ is the ray, then the points of the circle can be represented as $(x_0 + R\,\cos\theta, \,y_0 + R\,\sin\theta)$

The point inside or outside the circle can be represented as $M = (x_0 + \rho \,\cos\phi,\, y_0 + \rho\, \sin\phi)$

Then, the nearest point from $M$ on the circle is: $$ A = (x_0 + R \,\cos\phi,\, y_0 + R\, \sin\phi)$$

I guess this corresponds to your question ... which was not so clear

Answer 2

Three given points are vertices of a triangle $\triangle.$ The center of the circumcircle of $\triangle\;$ is the intersection of line bisectors of the sides of $\triangle.$ To find the center, it suffices to take two of the bisectors.
The point of the circle that is nearest to a given point lies on the diameter through the center and the given point. It is the red point at the picture.