Mathematics - Properties of Triangle Question with Solution | TestHub

Let a=BC, b=CA=2 and c=AB=1, so 1<a<3. External bisector lengths are 2bc sin(A/2)/|b-c|=4sin(A/2) and 2ab sin(C/2)/|a-b|=4a sin(C/2)/|a-2|. Their equality and the half-angle identities ${\sin }^2\left(A/2\right)$=($a^2$-1)/8 and ${\sin }^2\left(C/2\right)$=(3-a)(a-1)/(8a) give (a+1)(a-2)²=a(3-a) after squaring and dividing by a-1>0. Thus (a-1)($a^2$-a-4)=0. Since a≠1, a=(1+$\sqrt{17}$)/2, option C.