Disclaimer: The solutions we've shared are just one exciting approach, and there are surely many other wonderful methods out there. We’d love to hear your alternative solutions in the community thread below, so let's keep the creativity flowing!

Let $R$ and $T$ be on $BC$ and $DA$ respectively such that 

$PR\perp BC,~PT\perp DA$

So, $CR=DT$ and $AT=BR$

\[AP^2+CP^2=(AT^2+PT^2)+(PR^2+CR^2)=(BR^2+PR^2)+(PT^2+DT^2)=BP^2+DP^2\]

Now, $CP=CD=AB$. 

So,

$\implies 2(DP^2+PB^2)=DP^2+PB^2+AP^2+PC^2=PC^2+AB^2+DP^2$

$\implies DP^2=2(AB^2-PB^2)=2AP^2\implies DP=\sqrt{2}AP$

As wanted, $DP=\sqrt{2}AP$.