Hi everyone,
I have been reading a book about something called "Visual Calculus" which aims to solve problems that are traditionally solved with calculus through geometric methods instead. In one of the examples is the above image which I have included for reference.
The general idea is that the longest magenta line is a line tangent to the point $(x,x^3)$, and intersects the points $(x_{int},0)$ and $(0,y_{int})$, the $x$ and $y$ intercepts respectively. Also, the distance between $(x_{int},0)$ and $(0,y_{int})$ is three times the distance between $(x^3,x)$ and $(x_{int},0)$.
For the general case, the claim the mathematician makes in his book is that for any function $x^r$, the distance between the $x$ and $y$ intercept of a line tangent to the point $(x^r,x)$ will always be $r$ times the distance between $(x_{int},0)$ and $(x,x^r)$.
My question is how do you prove this? I have tried using the Pythagorean Theorem and the Distance Formula but the algebra involved with finding the scaling factor is extremely painful. Is there any other way to go about it?
Thank you!