If `r_1=r_2+r_3+r` prove that the triangle is right angled .

Explanation

We have `r_1=r_2+r_3 +r`
`rArr (Delta)/(s-a)-(Delta)/(s)=Delta/(s-b)+Delta/(s-c) rArr (s-s+a)/(s(s-a))=(s-c+s-b)/((s-b)(s-c))`
`rArr (a)/(s(s-a))=(2s-(b+c))/((s-b)(s-c) )" " {as,2s = a+b+c)`
`(a)/(s(s-a))=(a)/((s-b)(s-c))rArr s^2-(b+C)s+bc=s^2-as`
`rArr s(-a+b+c) rArr ((b+c-a)(a+b+c))/2=bc`
`rArr (b+c)^2-(a)^2=2bc " "rArr b^2+C^2+2bc-a^2=2bc`
`rArr b^(2)+c^(2)=a^(2)`
`therefore angle A= 90^(@)`