In ABC , if (r₂-r₁ ) (r₃-r₁ )=2 r₂ r₃ then 2(r+R)=

Step-by-step solution

aligned & Given that (r₂-r₁ ) (r₃-r₁ )=2 r₂ r₃ & (1- r₁ r₂ ) (1- r₁ r₃ )=2 & (1- s-a s-b ) (1- s-a s-c )=2 & (1- s-b s-a ) (1- s-c s-a )=2 aligned aligned & (b-a)(c-a)=2(s-a)^2 & (b-a)(c-a)= 1 2 (b+c-a)^2 b^2+c^2=a^2 aligned Hence the triangle is right-angled triangle with A=90^ We know that, aligned & a A = b B = c C =2 R & a=2 R A=2 R & b=2 R B and C=2 R C aligned We know that aligned & r=(s-a) A 2 =s-a= b+c-a 2 2 r=b+c-a & 2(r+R)=b+c-a+a=b+c aligned aligned & =2 R B+2 R C=2 R (2 B+C 2 B-C 2 ) & =4 R (45^ ) B-C 2