Given a circle $M$ of known center and radius and a right-angled triangle $JKL$ with one vertex (J) at the circle's center and another vertex (K) at the circle's boundary, find the cartesian coordinates of the third vertex (L) (with respect to the center of the circle). The three sides of the triangle JKL are given, where one of the sides (A) is a radius of the circle. The coordinates for the vertex K are not known.
Background: I intend to move a robotic arm based on x, and y movements to the center of a cylinder, these calculations are needed to determine that.
Let $r$ be the length of $JK,$ $c$ the length of $JL.$ We know the following:
Given any point $K$ on the outer circle, there are two lines passing through $K$ that are tangent to the inner circle, and either of these two points of tangency is a possible location for $L.$ Thus, $K$ could be any point on the outer circle, and even if we know for sure where $K$ is, we can only narrow possible positions for $L$ down to two distinct locations.
On the other hand, given any point $L$ on the inner circle, the tangent line to the inner circle at $L$ intersects the outer circle at exactly two points, and either of these two points of intersection is a possible location for $K.$ Thus, $L$ can theoretically be at any point on the inner circle, and even if we knew exactly where $L$ was, we could only narrow possible locations for $K$ down to two distinct locations.
In conclusion, we need more information to determine the location of $L.$