INMO 2007 Problem 5

Let $A B C$ be a triangle in which $A B=A C$. Let $D$ be the mid-point of $B C$ and $P$ be a point on $A D$. Suppose $E$ is the foot of perpendicular from $P$ on $A C$. If $\frac{A P}{P D}=\frac{B P}{P E}=\lambda, \frac{B D}{A D}=m$ and $z=m^{2}(1+\lambda)$, prove that

$$z^{2}-\left(\lambda^{3}-\lambda^{2}-2\right) z+1=0$$

Hence show that $\lambda \geq 2$ and $\lambda=2$ if and only if $A B C$ is equilateral.