A tangent line to a circle can only be formed if the line is perpendicular to the radius at the point of tangency. If the line is required to pass through a fixed point outside the circle, this limits the tangent lines to two as seen below. The moment segment [math]d[/math] is equal or less than the radius, no tangent lines can be formed anymore. This only happens when the fixed point is inside the circle. The slider [math]a[/math] changes the position of the point. Adjust this manually to see various scenarios.