Could you explain how the equation
$\hat{t}=-\frac{R-r}{r}t.$
can be derived from
$tR=(t-\hat{t})r$
step by step?
I'm trying to understand the mathematical aspect of Spirograph, and I found a quite nice explanation on Wikipedia (https://en.wikipedia.org/wiki/Spirograph), but only this part doesn't make sense to me.
Thank you in advance,