4A. At the intersection $O$ of the diagonals of the square $A B C D$, an arbitrary line $p$ is drawn, which does not pass through the vertices of the square. Find the sum of the squares of the distances from each vertex to the line $p$, if the length of the side of the square is $1 \mathrm{~m}$.
Solution. From the congruence of triangles $A O M$ and $C O R$ it follows that $\overline{A M}=\overline{C R}=x$, and from the congruence of triangles $B P O$ and $D N O$ it follows that $\overline{B P}=\overline{D N}=y . \quad$ From the congruence again, of right triangles $A M O$ and $B P O$ ( $\overline{A O}=\overline{B O}$, $\measuredangle A O M=\measuredangle O B P$ angles with mutually perpendicular sides) it follows that $\overline{O P}=\overline{A M}=x$. From the right triangle $O P B$ we get $\overline{O P}^{2}+\overline{B P}^{2}=\overline{B O}^{2}$, that is $x^{2}+y^{2}=\frac{1}{2}$. From this we have

$$
\overline{A M}^{2}+\overline{B P}^{2}+\overline{C R}^{2}+\overline{D N}^{2}=2 x^{2}+2 y^{2}=1
$$

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