The relation between the radiuses...

Question

Find $\frac{R}{r}$ where $R$ is the radius of the circumscribed circle of a trapezoid and $r$ is the radius of the inscribed circle of this trapezoid.

Thank you!

Answer 1

I'm assuming that you are looking for an isosceles bicentric trapezoid where the outcenter lies on the edge $AD$, since your figure looks that way. In that case you get

$$\frac Rr=1+\sqrt2$$

You can deduce this by starting with a right isosceles triangle:

$$A=(1,0)\qquad D=(0,1)\qquad E=(0,0)$$

Then the outcircle will have radius

$$R=\frac1{\sqrt2}$$

and the incircle will have radius

$$r=1-\frac1{\sqrt2}$$

which you can check by verifying that the incircle center $I=(r,r)$ has distance $r$ from the outcircle center $O=(\frac12,\frac12)$. You can turn this triangle into an isosceles trapezoid by cutting away the tip, i.e. by intersecting $OE$ with the incircle, resulting in point $F$, and then constructing a line through $F$ parallel to $AD$. But since you already have the circles, you can simply skip this and instead compute your fraction.