Mathematics - Straight Lines Question with Solution | TestHub

Reflection of A in the internal bisector of B is the point $B^{\prime }$ on line BC with $B{B}^{\prime }=BA=c$; $P_1$ is the midpoint of $A{B}^{\prime }$. Similarly $P_3$ is the midpoint of $A{C}^{\prime }$ with $C^{\prime }$ on line $\mathrm{BC},C{C}^{\prime }=CA=b$. Hence $P_1{P}_3=\frac{1}{2}{B}^{\prime }{C}^{\prime }=\frac{1}{2}(b+c-a)$. The condition $P_1{P}_3=\frac{1}{2}BC$ gives $b+c=2a$, which holds for many triangles (e.g. sides 4, 6, 5), not only equilateral ones. Hence nothing can be said.