The angle a tangent line forms with the circumference of a circle is is least possible angle made with a straight line from that point of tangent (tangent line EA in the image). This is demonstrated in Book III, proposition 16 of Euclid’s Elements.
The angle formed by the tangent is also known as a cornicular or horn angle
While it is possible to bisect any angle formed by two straight lines, the angle formed by the tangent line and a circumference cannot be bisected. It cannot be divided whatsoever. Though a curved line can be interposed between the vertex (such as a hyperbola), no rectilinear angle, no matter how small can be interposed.
But don’t you just want to bisect it? Don’t you think you might perchance fit some minuscule angle in there?
One way of visualizing the absolute nature of the tangent line is to picture what happens if one were to tilt it. The tangent would then become obtuse on one side of the circle, but on the other, no matter how little it is tilted, it would cease to form an angle. Rather, it would cut the circumference.
Imagine if one were to take a perfectly straight wooden plank and place it on a perfectly round basket ball. If one tilts it, it doesn’t change the angle it has to the ball, it changes its position--it rests upon a new point! This is the very reason we cannot move a ball on a flat surface without rolling it forward.
We are fooled because as the circle makes its arc, the space between the tangent line and the circle opens up–there looks to be space between tangent point which can be bisected. But there are an infinite number of points between the point of contact and the next point of the circle’s circumference. There is no next point, but always another prior to the point selected. And the point prior always has a smaller angular relationship with the tangent.