Problem: A circle of radius r is inscribed in a right isosceles triangle, and a circle of radius R is circumscribed about the triangle. Then rR​ equals
Answer Choices:
A. 1+2​
B. 22+2​​
C. 22​−1​
D. 21+2​​
E. 2(2−2​)
Solution:
In the adjoining figure. △ABC18 a right isosceles triangle, with ∠BAC=90∘ and AB=AC, Inscribed in a circle with center O and radius R The line segment AO has length R and bisects line segment BC and ∠BAC. A circle with center O′ lying on AO and radius r is inscribed in △ABC. The sides AB and AC are tangent to the inscribed circle with points of tangency T and T′, respectively. Since △ATO′ has angles 45∘−45∘−90∘ and O′T−r, AT=r and O′A=r2​, then R=r+r2​ and R/r=1+2​.
