JMO 1994 Problem 3

Let $P_0$ be a point in the plane of triangle $A_0A_1A_2$. Define $P_i\ (i=1,\ldots ,6)$ inductively as the point symmetric to $P_{i-1}$ with respect to $A_k$, where $k$ is the remainder when $i$ is divided by $3$.

a) Prove that $P_6\equiv P_1$. b) Find the locus of points $P_0$ for which $P_iP_{i+1}$ does not meet the interior of $\triangle A_0A_1A_2$ for $0\le i\le 5$.