The angles of a right - angled triangle are in AP. The ratio of the inradius and the perimeter is
(α−β)+α+(α+β)=1800 and α+β=900∴The angles are300,600,900∴asin300=bsin600=csin900=2Ror a=R,b=√3R,c=2Rr=Δs=12R.√3R12(R+√3R+2R)=√3R3+√3and 2s=R+√3R+2R=(3+√3)R∴r2s=√3R3+√3×13+√3R=1(3+√3)(√3+1)=16+4√3=12√3(2+√3)=2−√32√3