I have a question about hyperbolas as follows.
Given this basic diagram of a hyperbola, with points $a, b, c$ (and their negatives respectively), how can I prove the fundamental formula $a^2 + b ^2 = c^2 $ geometrically (in terms of $a, b,$ and $c$) and be able to explain it to my math teacher and class (high school sophomores in PreCalculus). 1 My difficulty lies in how the line from $(0,b)$ to $(a,0)$ shown in light blue is the same as $(c,0)$ also shown in light blue.
Best, A PreCalculus Student