I am missing some fundamental hyperbola geometry-related concepts probably and I cannot get more insight from the websites, so I'd like to ask the following from this forum.
Let A = x(A), B = Length(B), and C = y(C)
Is there any neat way to explain, why the run and the rise line B/A (x = yB/A) intersecting hyperbola (y^2-x^2 = A^2) and the tangent of the circle with the radius A meet at the Y-coordinate on point C? Should be also said that the tangent crosses the x-axis in x=B.
I can see how the math equates, but I feel I have not understood why seemingly different geometric constructions yield the same position on the Y-axis. Is this due to some explicit mathematical theorem?
Below is the construction. Pardon me for not providing more code at this point:
The line and the hyperbola is solved in the picture C = AB/sqrt(B^2-A^2).