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 $\frac{RT}{RQ}$?


Circle has equation $x^2+y^2=4$
$U(1;\;-2)$
The equation of a general line passing through $U$ is
$mx-y-m-2=0$
This line is tangent to the circle if its distance from the center is equal to the radius, that is
$$\frac{|-m-2|}{\sqrt{m^2+1}}=2\to m^2+4m+4=4m^2+4\to m_1=0;\;m_2=\frac{4}{3}$$
One tangent is the line containing $RS$ the other is the line $UT$
$y=\dfrac{4}{3}x-\dfrac{10}{3}$
which for $x=2$ gives $y=-\dfrac{2}{3}$ and $RT=2-\dfrac{2}{3}=\dfrac{4}{3}$
This means $\dfrac{RT}{RQ}=\dfrac{\frac{4}{3}}{4}=\dfrac{1}{3}$
Hope this can be useful

