Let ABC be a triangle in which AB\equal{}AC. Let D be the midpoint of BC and P be a point on AD. Suppose E is the foot of perpendicular from P on AC. Define
\frac{AP}{PD}\equal{}\frac{BP}{PE}\equal{}\lambda , \ \ \ \frac{BD}{AD}\equal{}m , \ \ \ z\equal{}m^2(1\plus{}\lambda)
Prove that
z^2 \minus{} (\lambda^3 \minus{} \lambda^2 \minus{} 2)z \plus{} 1 \equal{} 0
Hence show that λ≥2 and \lambda \equal{} 2 if and only if ABC is equilateral. inequalitiesgeometry unsolvedgeometry