Can you please help me solve this really difficult problem: Find 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 very much for your trying to help, in advance !
I believe your answer is given on these links: Isosceles tangential trapezoid and Bicentric quadrilateral.
UPDATE :
This journal can answer your question, just click and it will automatically download the journal. I found this journal in the link that I gave to you. I hope this help.
Your question can be solved by using Fuss' theorem (you can see this theorem on the links that I gave to you). Fuss' theorem gives a relation between the inradius $r$, the circumradius $R$ and the distance $x$ between the center of the inner circle and center of the outer circle, for any bicentric quadrilateral (trapezoid is included). The relation is $$ \frac{1}{(R-x)^2}+\frac{1}{(R+x)^2}=\frac{1}{r^2}. $$ In this case, using your picture, we have $x=r$. Therefore $$ \frac{1}{(R-r)^2}+\frac{1}{(R+r)^2}=\frac{1}{r^2}.\tag1 $$ By using equation $(1)$, I think it is not difficult to obtain that: $$ \frac{R}{r}=\sqrt{2+\sqrt{5}}. $$
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