Found problems: 467
Consider in a square $[ABCD]$ a point $E$ on the side $AB$, different from $A$ and $B$. On the side $BC$ consider the point $F$ such that $\angle AED = \angle DEF$ . Prove that $EF = AE + FC$.
Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$?
Mikhail Evdokimov
We consider a square $ABCD$ and a point $E$ on the segment $CD$. The bisector of $\angle EAB$ cuts the segment $BC$ in $F$. Prove that $BF + DE = AE$.
Let $K, L,M,N$ designate the centers of the squares erected on the four sides (outside) of a rhombus. Prove that the polygon $KLMN$ is a square.
Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than $180^o$.
Let $ABCD$ be a square of side paper $10$ and $P$ a point on side $BC$. By folding the paper along the $AP$ line, point $B$ determines the point $Q$, as seen in the figure. The line $PQ$ cuts the side $CD$ at $R$. Calculate the perimeter of the triangle $ PCR$
[img]https://3.bp.blogspot.com/-ZSyCUznwutE/XNY7cz7reQI/AAAAAAAAKLc/XqgQnjm8DQYq6Q7fmCAKJwKt3ihoL8AuQCK4BGAYYCw/s400/may%2B2013%2Bl1.png[/img]
Inside the square $ABCD$ there is the square $A'B' C'D'$ so that the segments $AA', BB', CC'$ and $DD'$ do not intersect each other neither the sides of the smaller square (the sides of the larger and the smaller square do not need to be parallel). Prove that the sum of areas of the quadrangles $AA'B' B$ and $CC'D'D$ is equal to the sum of areas of the quadrangles $BB'C'C$ and $DD'A'A$.
The square $ABCD$ is inscribed in a circle. Points $E$ and $F$ are located on the side of the square, and points $G$ and $H$ are located on the smaller arc $AB$ of the circle so that the $EFGH$ is a square. Find the area ratio of these squares.
Four masts stand on a flat horizontal piece of land at the vertices of a square $ABCD$. The height of the mast on $A$ is $7$ meters, of the mast on $B$ $13$ meters, and of the mast on $C$ $15$ meters. Within the square there is a point $P$ on the ground equidistant from each of the tops of these three masts.
(a) What length must the sides of the square be at least for this to be possible?
(b) The distance from $P$ to the top of the mast on $D$ is equal to the distance from$ P$ to each of the tops of the three other masts. Calculate the height of the mast at $D$.
Let $ABCD$ be a square and let $M$ be the midpoint of $BC$. Let $C ^ \prime$ be the reflection of $C$ wrt to $DM$. The parallel to $AB$ passing through $C ^ \prime$ intersects $AD$ at $R$ and $BC$ at $S$. Show that $$\frac {RC ^ \prime} {C ^\prime S} = \frac {3} {2}$$
Points $A, B$ and $C$ lie on the same line so that $CA = AB$. Square $ABDE$ and the equilateral triangle $CFA$, are constructed on the same side of line $CB$. Find the acute angle between straight lines $CE$ and $BF$.
A triangle with a parallelogram inside was placed in a square. Prove that the area of a parallelogram is not more than a quarter of a square.
A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.
Given in the plane the square $ABCD$, the square $A_1B_1C_1D_1$, smaller than the first, and a quadrilateral $PQRS$ that satisfy the following conditions
$\bullet$ $ABCD$ and $A_1B_1C_1D_1$ have a common center and respectively parallel sides.
$\bullet$$P$, $Q$, $R$, $S$ belong one to each side of the square $ABCD$.
$\bullet$ $A_1$, $B_1$, $C_1$, $D_1$ belong one to each side of the quadrilateral $PQRS$.
Prove that $PQRS$ is a square.
Let $ABCD$ be a square and let point $F$ be any point on side $BC$. Let the line perpendicular to $DF$, that passes through $B$, intersect line $DC$ at $Q$. What is value of $\angle FQC$?
Two lines divide a square into $4$ figures of the same area. Prove that all these figures are congruent.
Two squares with side $12$ lie exactly on top of each other.
One square is rotated around a corner point through an angle of $30$ degrees relative to the other square.
Determine the area of the common piece of the two squares.
[asy]
unitsize (2 cm);
pair A, B, C, D, Bp, Cp, Dp, P;
A = (0,0);
B = (-1,0);
C = (-1,1);
D = (0,1);
Bp = rotate(-30)*(B);
Cp = rotate(-30)*(C);
Dp = rotate(-30)*(D);
P = extension(C, D, Bp, Cp);
fill(A--Bp--P--D--cycle, gray(0.8));
draw(A--B--C--D--cycle);
draw(A--Bp--Cp--Dp--cycle);
label("$30^\circ$", (-0.5,0.1), fontsize(10));
[/asy]
The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?
Let $ABCD$ be a quadrilateral that can be inscribed in a circle whose diagonals are perpendicular. Denote by $P$ and $Q$ the feet of the perpendiculars through $D$ and $C$ respectively on the line $AB$, $X$ is the intersection point of the lines $AC$ and $DP$, $Y$ is the intersection point of the lines $BD$ and $CQ$. Show that $XY CD$ is a rhombus.
Let $ABCD$ be a parallelogram of center $O$. Points $M$ and $N$ are the midpoints of $BO$ and $CD$, respectively. Prove that if the triangles $ABC$ and $AMN$ are similar, then $ABCD$ is a square.
How many different real numbers $x$ satisfy the equation $$(x^2-5)^2=16?$$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8$
Given a square $ABCD$, point $E$ is the midpoint of $AD$. Let $F$ be the foot of the perpendicular drawn from point $B$ on $EC$. Point $K$ on $AB$ is such that $\angle DFK = 90^o$. The point $N$ on the $CE$ is such that $\angle NKB = 90^o$. Prove that the point $N$ lies on the segment $BD$.
(Matvii Kurskyi)
[img]https://cdn.artofproblemsolving.com/attachments/4/2/d42b8c8117ec1d5e5c5b981904779b156fce93.png[/img]
Show that a circle inscribed in a square has a larger perimeter than any other ellipse inscribed in the square.
There is a square $ABCD$ in the plane with $|AB|=a$. Determine the smallest possible radius value of three equal circles to cover a given square.
Let $E$ be an arbitrary point different from $A$ and $B$ on the side $AB$ of a square $ABCD$, and let $F$ and $G$ be points on the segment $CE$ so that $BF$ and $DG$ are perpendicular to $CE$. Prove that $DF = AG$.