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`
or `r_(1) - r = r_(2) + r_(3)`
`rArr (Delta)/(s -a) -(Delta)/(s) = (Delta)/(s -b) + (Delta)/(s -c)`
or `(Delta a)/(s(s -a)) = (Delta (2s -b -c))/((s-b) (s-c)) = (Delta a)/((s-b) (s-c))`
or `s(s -a) = (s -b) (s -c)`
or `s^(2) - sa = s^(2) - (b + c) s + bc`
or `2s (b + c-a) = 2bc`
or `(a + b + c) (b + c-a) = 2bc`
or `(b + c)^(2) - a^(2) = 2bc`
or `b^(2) + c^(2) = a^(2)`
Hence, the triangle is right angled