Question

what is the ratio of inradius and circumradius of right angled triangle

Answer 1

Let ABC  a right angled triangle right angle at C then
 inradius r=(s-a)tan(A/2)=(s-b)tan(B/2)=(s-c)tan(C/2)  where 's' is the semi perimeter 
 since  ∠C=90 degrees 
r=s-c        and R=abc/4Δ      
Δ=1/2 ab   
∴  2R=c
∴r+2R=c