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I have a point '$a$' with known coordinates, from which I have drawn a tangent to a circle with center '$c$' which is also known. What is the best way of finding the coordinates of point '$b$', the touching point between the tangent and the circle? Here is a diagram:



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up vote 2 down vote accepted

If numeric precision is not a problem you can use trigonometry. Find ab from the Pythagorean theorem, find the angle at a from bc/ac, calculate the angle of ac relative to the horizontal, add the triangle angle and get the slope of ab. If the angle gets very small, you may want to use the small angle approximations.

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You have $(x-x_c)^2+(y-y_c)^2=r^2$ for the circle. You also have $(y-y_a)=m(x-x_a)$. $m=(x_c-x_a)/(y_c-y_a)$. Take the second equation, solve for $y$, plot it into the first equation, and solve for $x$.

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