Let $ABCD$ be a square and let $X$ be any point on side $BC$ between $B$ and $C$. Let $Y$ be the point on line $CD$ such that $BX = YD$ and $D$ is between $C$ and $Y$ . Prove that the midpoint of $XY$ lies on diagonal $BD$.

In a plane, a square $ABCD$ and a point $P$ located inside it are given. Let a point $ Q$ pass through all sides of the square. Describe the set of all those points $R$ in for which the triangle $PQR$ is equilateral.

The segment $AB$ divides the square into two parts, in each of which a circle can be inscribed. The radii of these circles are equal to $r_1$ and $r_2$ respectively, where $r_1> r_2$. Find the length of $AB$.

In the center of the square of side $1$ shown in the figure is an ant. At one point the ant starts walking until it touches the left side $(a)$, then continues walking until it reaches the bottom side $(b)$, and finally returns to the starting point. Show that, regardless of the path followed by the ant, the distance it travels is greater than the square root of $2$. [asy] unitsize(2 cm); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); label("$a$", (0,0.5), W); label("$b$", (0.5,0), S); dot((0.5,0.5)); [/asy]

We are given two concentric circles, Construct a square whose two vertices lie on one circle and the other two on the other circle.

Let $E$ be an arbitrary point on the side $BC$ of the square $ABCD$. Prove that the inscribed circles of triangles $ABE$, $CDE$, $ADE$ have a common tangent.

Three circles of radius $1$ are inscribed in a square of side length $s$, such that the circles do not overlap or coincide with each other. What is the minimum $s$ where such a configuration is possible?

The edges of the square in the figure have length $1$. Find the area of the marked region in terms of $a$, where $0 \le a \le 1$. [img]https://cdn.artofproblemsolving.com/attachments/2/2/f2b6ca973f66c50e39124913b3acb56feff8bb.png[/img]

Let the diagonals of the square $ABCD$ intersect at $S$ and let $P$ be the midpoint of $AB$. Let $M$ be the intersection of $AC$ and $PD$ and $N$ the intersection of $BD$ and $PC$. A circle is incribed in the quadrilateral $PMSN$. Prove that the radius of the circle is $MP- MS$.

The side $AD$ of a square $ABCD$ is the base of an obtuse-angled isosceles triangle $AED$ with vertex $E$ lying inside the square. Let $AF$ be a diameter of the circumcircle of this triangle, and $G$ be a point on $CD$ such that $CG = DF$. Prove that angle $BGE$ is less than half of angle $AED$.

The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square. [img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]

Consider the square $ABCD$ and the point $E$ on the side $AB$. The line $DE$ intersects the line $BC$ at point $F$, and the line $CE$ intersects the line $AF$ at point $G$. Prove that the lines $BG$ and $DF$ are perpendicular.

In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$. [img]https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png[/img]

Can the distances from a certain point on the plane to the vertices of a certain square be equal to $1, 4, 7$, and $8$ ?

In the square $ABCD$ denote by $M$ the midpoint of the side $[AB]$, with $P$ the projection of point $B$ on the line $CM$ and with $N$ the midpoint of the segment $[CP]$, Bisector of the angle $DAN$ intersects the line $DP$ at point $Q$. Show that the quadrilateral $BMQN$ is a parallelogram.

Suppose we have a square $ABCD$ and a point $S$ in the interior of this square. Under homothety with centre $S$ and ratio of magnification $k > 1$, this square becomes another square $A'B'C'D'$. Prove that the sum of the areas of the two quadrilaterals $A'ABB'$ and $C'CDD'$ are equal to the sum of the areas of the two quadrilaterals $B'BCC'$ and $D'DAA'$. [asy] unitsize(3 cm); pair[] A, B, C, D; pair S; A[1] = (0,1); B[1] = (0,0); C[1] = (1,0); D[1] = (1,1); S = (0.3,0.6); A[0] = interp(S,A[1],2/3); B[0] = interp(S,B[1],2/3); C[0] = interp(S,C[1],2/3); D[0] = interp(S,D[1],2/3); draw(A[0]--B[0]--C[0]--D[0]--cycle); draw(A[1]--B[1]--C[1]--D[1]--cycle); draw(A[1]--S, dashed); draw(B[1]--S, dashed); draw(C[1]--S, dashed); draw(D[1]--S, dashed); dot("$A$", A[0], N); dot("$B$", B[0], SE); dot("$C$", C[0], SW); dot("$D$", D[0], SE); dot("$A'$", A[1], NW); dot("$B'$", B[1], SW); dot("$C'$", C[1], SE); dot("$D'$", D[1], NE); dot("$S$", S, dir(270)); [/asy]