If an exradius of a triangle is the sum of the two other exradii and the inradius, then the triangle is ...

Question

In a triangle $ABC$, if an exradius is the sum of the two other exradii and the inradius, the triangle is

1. equilateral
2. isosceles
3. scalene
4. right-angled

I've tried multiple times, but this is the most simplified expression I've got: $$r_1 = r_2 + r_3 + r$$ $$sX\tan\frac A2=sX\tan\frac B2+sX\tan\frac C2+(s-a)\tan\frac A2$$ $$sX\tan\frac A2=sX\tan\frac B2+sX\tan\frac C2+sX\tan\frac A2-aX\tan\frac A2$$ $$aX\tan\frac A2=sX\tan\frac B2+sX\tan\frac C2$$ I could not go further.