Mathematics - Properties of Triangle Question with Solution | TestHub

Coefficients add to zero, so 1 is a root; equality of roots makes 1 a repeated root. Thus c(a-b)=a(b-c), or b=2ac/(a+c), so a,b,c are in HP. As s=(a+b+c)/2, the values (s-a)/a,(s-b)/b,(s-c)/c are in AP. Multiplication by the same positive abc/[(s-a)(s-b)(s-c)] shows bc/[(s-b)(s-c)], ca/[(s-c)(s-a)], ab/[(s-a)(s-b)] are in AP. Their reciprocals are ${\sin }^2\left(A/2\right)$,${\sin }^2\left(B/2\right)$,${\sin }^2\left(C/2\right)$, since ${\sin }^2\left(A/2\right)$=(s-b)(s-c)/(bc). Hence HP.