Ratio of an inscribed circle's tangent to original square In the diagram, the circle is inscribed within square $PQRS,$ $\overline{UT}$ is tangent to the circle, and $RU$ is $\frac{1}{4}$ of $RS.$ What is $\frac{RT}{RQ}$?


Suppose that the circle touches $RS$, $TU$ and $RQ$ at $X$, $Y$ and $Z$ respectively.
If $RQ=4$, $RU=1$. Let $RT=x$.
Then $TU=TY+YU=TZ+XU=2-x+1$.
So $1^2+x^2=(3-x)^2$ and thus $1+x^2=9-6x+x^2$.
$\displaystyle x=\frac{4}{3}$.
$\displaystyle \frac{RT}{RQ}=\frac{1}{3}$.
