## Task A-1.3.

Let $A B C D$ be a rectangle and $k$ a circle with its center at the center of the rectangle. The circle $k$ intersects side $\overline{A B}$ at points $K$ and $L$, and side $\overline{C D}$ at points $M$ and $N$, and $L N \perp A C$. If the midpoint of segment $\overline{A N}$ lies on the circle $k$, what is the ratio of the lengths of the sides of the given rectangle?
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_77c5fa69d8ea891e3e07g-03.jpg?height=477&width=739&top_left_y=624&top_left_x=664)

Let $S$ be the center of the circle $k$ and the rectangle $ABCD$, and let $P$ be the midpoint of the segment $\overline{AN}$. Let $r$ be the radius of the circle $k$.

Clearly, $\overline{NL}$ is the diameter of the circle $k$. (See the note.)

Since $LN \perp AC$, triangle $ASN$ is a right triangle.

The segment $\overline{PS}$ is the median of this triangle, so $|AP| = |PN| = |PS|$.

From $|PN| = |PS| = r = |NS|$, it follows that triangle $NPS$ is equilateral.

The segment $\overline{PS}$ is the median of triangle $ALN$ (since $|AP| = |NP|$ and $|LS| = |SN|$),

so $|AL| = 2|PS| = 2r$ and triangle $ANL$ is also equilateral.

Triangle $ADN$ is a right triangle, its hypotenuse $\overline{AN}$ is of length $2r$ and $\varangle NAD = 90^{\circ} - \varangle LAN = 30^{\circ}$.

Therefore, $|DN| = \frac{1}{2}|AN| = r$ and $|DA| = |DN| \sqrt{3} = r \sqrt{3}$.

Quadrilateral $ALCN$ is a rhombus (its diagonals are perpendicular and bisect each other), so $|NC| = |AN| = 2r$.

Thus, $|DC| = |DN| + |NC| = r + 2r = 3r$.

The desired ratio is $|AD|:|CD| = \frac{r \sqrt{3}}{3r} = \frac{1}{\sqrt{3}}$. (Or $|AB|:|AD| = \sqrt{3}$.)

Note. Proof that $\overline{NL}$ is the diameter of the circle $k$:

Let $G$ and $H$ be the midpoints of the sides $\overline{AB}$ and $\overline{CD}$. Triangles $SGL$ and $SHN$ are congruent ($\varangle SGL = \varangle SHN = 90^{\circ}$, $|SL| = |SN| = r$, $|SG| = |SH| = \frac{1}{2}|BC|$), so $\varangle GSL = \varangle HSN$. We conclude that $NL$ passes through $S$.

It is not required that students prove this and similar statements that hold due to symmetry!