The numbers $a$, $b$, $c$, $d$ satisfy $0<a \leq b \leq d \leq c$ and ${a+c=b+d}$. Prove that for every internal point $P$ of a segment with length $a$ this segment is a side of a circumscribed quadrilateral with consecutive sides $a$, $b$, $c$, $d$, such that its incircle contains~$P$.