2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.

There are rhombus $ABCD$ and circle $\Gamma_B$, which is centred at $B$ and has radius $BC$, and circle $\Gamma_C$, which is centred at $C$ and has radius $BC$. Circles $\Gamma_B$ and $\Gamma_C$ intersect at point $E$. The line $ED$ intersects $\Gamma_B$ at point $F$. Find all possible values of $\angle AFB$.

Four segments drawn from a given point inside a convex quadrilateral to its vertices, split the quadrilateral into four equal triangles. Can we assert that this quadrilateral is a rhombus?

A square sheet of paper $ABCD$ is so folded that $B$ falls on the mid point of $M$ of $CD$. Prove that the crease will divide $BC$ in the ration $5 : 3$.

We have a rhombus $ABCD$ with $\angle BAD=60^\circ$. We take points $F,H,G$ on the sides $AD,DC$ and the diagonal $AC$, respectively, such that $DFGH$ is a parallelogram. Prove that $BFH$ is equilateral.

Let $ABCD$ be rhombus, with $\angle ABC = 80^o$: Let $E$ be midpoint of $BC$ and $F$ be perpendicular projection of $A$ onto $DE$. Find the measure of $\angle DFC$ in degree.

Of a rhombus $ABCD$ we know the circumradius $R$ of $\Delta ABC$ and $r$ of $\Delta BCD$. Construct the rhombus.

On the sides $AD$ and $BC$ of a rhombus $ABCD$ we consider the points $M$ and $N$ respectively. The line $MC$ intersects the segment $BD$ in the point $T$, and the line $MN$ intersects the segment $BD$ in the point $U$. We denote by $Q$ the intersection between the line $CU$ and the side $AB$ and with $P$ the intersection point between the line $QT$ and the side $CD$. Prove that the triangles $QCP$ and $MCN$ have the same area.

i) $ABCD$ is a rhombus. A tangent to the inscribed circle meets $AB, DA, BC, CD$ at $M, N, P, Q$ respectively. Find a relationship between $BM$ and $DN$. ii) $ABCD$ is a rhombus and $P$ a point inside. The circles through $P$ with centers $A, B, C, D$ meet the four sides $AB, BC, CD, DA$ in eight points. Find a property of the resulting octagon. Use it to construct a regular octagon. iii) Rotate the figure about the line $AC$ to form a solid. State a similar result.

Rhombus $ ABCD$, with a side length $ 6$, is rolled to form a cylinder of volume $ 6$ by taping $ \overline{AB}$ to $ \overline{DC}.$ What is $ \sin(\angle ABC)$? $ \textbf{(A)}\ \frac {\pi}{9} \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {\pi}{6} \qquad \textbf{(D)}\ \frac {\pi}{4} \qquad \textbf{(E)}\ \frac {\sqrt3}{2}$

In the Cartesian plane $XOY$, there is a rhombus $ABCD$ whose side lengths are all $4$ and $|OB|=|OD|=6$, where $O$ is the origin. [b](1)[/b] Prove that $|OA|\cdot |OB|$ is a constant. [b](2)[/b] Find the locus of $C$ if $A$ is a point on the semicircle \[(x-2)^2+y^2=4 \quad (2\le x\le 4).\]

Let $ABC$ be a triangle with altitude $\overline{AE}$. The $A$-excircle touches $\overline{BC}$ at $D$, and intersects the circumcircle at two points $F$ and $G$. Prove that one can select points $V$ and $N$ on lines $DG$ and $DF$ such that quadrilateral $EVAN$ is a rhombus. [i]Danielle Wang and Evan Chen[/i]

Let be an isosceles trapezoid such that its smaller base is equal to its legs, and a rhombus that has each of its vertexes on a different side of the trapezoid. Prove that the smaller angles of the trapezoid are equal to the smaller ones of the rhombus. [i]Vlad Robu[/i]

Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .

Points $C_0$ and $B_0$ are the respective midpoints of sides $AB$ and $AC$ of a non-isosceles acute triangle $ABC$, $O$ is its circumscenter and $H$ is the orthocenter. Lines $BH$ and $OC_0$ intersect at $P$, while lines $CH$ and $OB_0$ intersect at $Q$. $OPHQ$ is rhombus. Prove that points $A$, $P$ and $Q$ are collinear. (Author: L. Emelyanov)

$ABCD$ forms a rhombus. $E$ is the intersection of $AC$ and $BD$. $F$ lie on $AD$ such that $EF$ is perpendicular to $FD$. Given $EF=2$ and $FD=1$. Find the area of the rhombus $ABCD$

Let $ABC$ be a triangle, $O$ its circumcenter, $S$ its centroid, and $H$ its orthocenter. Denote by $A_1, B_1$, and $C_1$ the centers of the circles circumscribed about the triangles $CHB, CHA$, and $AHB$, respectively. Prove that the triangle $ABC$ is congruent to the triangle $A_1B_1C_1$ and that the nine-point circle of $\triangle ABC$ is also the nine-point circle of $\triangle A_1B_1C_1$.

Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$

In the triangle $ABC$ the angle bisectors $AL$ and $BT$ are drawn, which intersect at the point $I$, and their extensions intersect the circle circumscribed around the triangle $ABC$ at the points $E$ and $D$ respectively. The segment $DE$ intersects the sides $AC$ and $BC$ at the points $F$ and $K$, respectively. Prove that: a) quadrilateral $IKCF$ is rhombus; b) the side of this rhombus is $\sqrt {DF \cdot EK}$. (Rozhkova Maria)

Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$.

Given a rhombus $ABCD$. A point $M$ is chosen on its side $BC$. The lines, which pass through $M$ and are perpendicular to $BD$ and $AC$, meet line $AD$ in points $P$ and $Q$ respectively. Suppose that the lines $PB,QC,AM$ have a common point. Find all possible values of a ratio $\frac{BM}{MC}$. [i]S. Berlov, F. Petrov, A. Akopyan[/i]

Four circles of radius $ 1$ are each tangent to two sides of a square and externally tangent to a circle of radius $ 2$, as shown. What is the area of the square? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); real h=3*sqrt(2)/2; pair O0=(0,0), O1=(h,h), O2=(-h,h), O3=(-h,-h), O4=(h,-h); pair X=O0+2*dir(30), Y=O2+dir(45); draw((-h-1,-h-1)--(-h-1,h+1)--(h+1,h+1)--(h+1,-h-1)--cycle); draw(Circle(O0,2)); draw(Circle(O1,1)); draw(Circle(O2,1)); draw(Circle(O3,1)); draw(Circle(O4,1)); draw(O0--X); draw(O2--Y); label("$2$",midpoint(O0--X),NW); label("$1$",midpoint(O2--Y),SE);[/asy]$ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 22 \plus{} 12\sqrt {2}\qquad \textbf{(C)}\ 16 \plus{} 16\sqrt {3}\qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 36 \plus{} 16\sqrt {2}$

Let ABC be a triangle, G its centroid, M the midpoint of AB, D the point on the line $AG$ such that $AG = GD, A \neq D$, E the point on the line $BG$ such that $BG = GE, B \neq E$. Show that the quadrilateral BDCM is cyclic if and only if $AD = BE$.

Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$? [asy] unitsize(1cm); draw((0,1)--(2,2)--(4,1)--(2,0)--cycle); dot("$A$",(0,1),W); dot("$D$",(2,2),N); dot("$C$",(4,1),E); dot("$B$",(2,0),S); [/asy] $\textbf{(A) } 60 \qquad\textbf{(B) } 90 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 120 \qquad\textbf{(E) } 144$

The sides of a rhombus have length $a$ and the area is $S$. What is the length of the shorter diagonal?