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Let diagonals intersect at right angles. Then we have 

with equality if $P$ lies on $\overline{AC}$ and $\overline{BD}$ respectively. Thus

Since we have the equality case, $\overline{AC} \perp \overline{BD}$ at point $P$ 

By the Pythagorean Theorem,

The perimeter of $ABCD$ is 

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