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There are 2 circles, the smaller one has its center on the bigger circles border, from that how can you calculate the coordinates the closest point on the smaller circle to the center of the bigger one.

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I want to find the coordinates of the closest point of the circle given that I have the height and width and radius of both circles and the position of both circles.

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  • $\begingroup$ Connect the two centers with a line, then intersect that line with the small circle. $\endgroup$
    – vadim123
    Commented Jul 14, 2014 at 18:48
  • $\begingroup$ @vadim123 maybe the question wasn't clear enough, I edited it, I want to find the position of the point in a coordinate system. $\endgroup$
    – Nick
    Commented Jul 14, 2014 at 18:52
  • $\begingroup$ You can follow my method algebraically. Which step do you have trouble with? $\endgroup$
    – vadim123
    Commented Jul 14, 2014 at 18:54

2 Answers 2

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If you are certain that the small circle has the centre on the larger circles border then find the tangent to the point on the small circle that connects the centre on the small circle to that of the centre of the larger circle.

After that we have the equation: X = (x-x1) & Y = (y-y1) From distance formula: (X^2 + Y^2)^1/2 = C where C is the radius of the larger circle.

Equate this equation with the tangent of the point on the small circle as mentioned earlier as a simultaneous equation and you will get both x and y coordinates of the closest point.

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If you know the radius of both circles, then the distance from the center of the larger circle to the closest point on the smaller circle is $r_1-r_2$, where $r_1$ is the radius of the larger circle and $r_2$ is the radius of the smaller. If, as it appears, the center of the smaller circle is also on the $x$-axis, then you have your answer, once you make it negative and plug it into an ordered pair. If not, then you need to use the trigonometric functions to find the coordinates of the center, and then solve for the coordinates of a point $r_1-r_2$ in the direction of the angle.

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