We know
#r_1=Delta/(s-a)#
#r_2=Delta/(s-b)#
#r_3=Delta/(s-c)#
#r=Delta/s#
Inserting these values in the given relation.
#r_1=r_2+r_3+r#
#=>r_1-r=r_2+r_3#
#=>Delta/(s-a)-Delta/s=Delta/(s-b)+Delta/(s-c)#
#=>1/(s-a)-1/s=1/(s-b)+1/(s-c)#
#=>2/(2s-2a)-2/(2s)=2/(2s-2b)+2/(2s-2c)#
#=>1/(b+c-a)-1/(b+c+a)=1/(c+a-b)+1/(a+b-c)#
#=>((b+c+a)-(b+c-a))/((b+c)^2-a^2)=(c+a-b+a+b-c)/(a^2-(b-c)^2)#
#=>(2a)/((b+c)^2-a^2)=(2a)/(a^2-(b-c)^2)#
#=>(b+c)^2-a^2=a^2-(b-c)^2#
#=>(b+c)^2+(b-c)^2=a^2+a^2#
#=>2b^2+2c^2=2a^2#
#=>b^2+c^2=a^2#
This means the triangle is right angled.
