[ Properties and characteristics of tangents ]  [ Pythagorean Theorem (direct and inverse) ]

Find the geometric locus of points from which tangents are drawn to a given circle, equal to a given segment.

#
Consider a right triangle with vertices at the considered point, the center of the given circle, and the point of tangency.

## Solution

Let the tangent line drawn from point $M$ to the given circle be $d$ (i.e., the segment with endpoints at point $M$ and the point of tangency is equal to the given length), and the radius of this circle be $r$. Then, point $M$ is at a distance of $\sqrt{r^{2}+d^{2}}$ from the center of the circle. Therefore, point $M$ is located on a circle with the same center as the given circle and a radius of $\sqrt{r^{2}+d^{2}}$.

On the other hand, if point $K$ lies on a circle with radius $\sqrt{r^{2}+d^{2}}$ and center at point $O$ (the center of the given circle), then the tangent line drawn from point $K$ to the given circle is equal to $d$.

![](https://cdn.mathpix.com/cropped/2024_05_06_633abcb7b0e2bd041761g-08.jpg?height=429&width=472&top_left_y=11&top_left_x=793)

## Answer

A circle concentric with the given one.