$\displaystyle \frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{rz}$ and p,q,r be in AP , then prove that x,y,z are in HP.
If $\displaystyle \frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{rz} = c$, then $\displaystyle a = x(cp+1) = y(cq+1) = z(cr+1)$, from which $\displaystyle \tfrac1x = \tfrac1a(cp+1)$, $\displaystyle \tfrac1y = \tfrac1a(cq+1)$, $\displaystyle \tfrac1z = \tfrac1a(cr+1)$. But if p, q and r are in AP then so are $\displaystyle \tfrac1a(cp+1)$, $\displaystyle \tfrac1a(cq+1)$ and $\displaystyle \tfrac1a(cr+1)$.