In a parallelogram $ABCD$ with the side ratio $AB : BC = 2 : \sqrt 3$ the normal through $D$ to $AC$ and the normal through $C$ to $AB$ intersects in the point $E$ on the line $AB$. What is the relationship between the lengths of the diagonals $AC$ and $BD$?

Let $ABCD$ be a parallelogram. Let $M$ be the midpoint of the arc $AC$ containing $B$ of the circumcircle of $ABC$ . Let $E$ be a point on segment $AD$ and $F$ a point on segment $CD$ such that $ME=MD=MF$. Show that $BMEF$ is cyclic. [i]Proposed by A. Tereshin[/i]

Let $ABCD$ be a parallelogram and let $E$ be a point in the plane such that $AE\perp AB$ and $BC\perp EC$. Show that either $\angle AED=\angle BEC$ or $\angle AED+\angle BEC=180^\circ$.

Let $ABCD$ be a parallelogram (non-rectangle) and $\Gamma$ is the circumcircle of $\triangle ABD$. The points $E$ and $F$ are the intersections of the lines $BC$ and $DC$ with $\Gamma$ respectively. Define $P=ED\cap BA$, $Q=FB\cap DA$ and $R=PQ\cap CA$. Prove that $$\frac{PR}{RQ}=(\frac{BC}{CD})^2$$

Let $r,R$ and $r_a$ be the radii of the incircle, circumcircle and A-excircle of the triangle $ABC$ with $AC>AB$, respectively. $I,O$ and $J_A$ are the centers of these circles, respectively. Let incircle touches the $BC$ at $D$, for a point $E \in (BD)$ the condition $A(IEJ_A)=2A(IEO)$ holds. Prove that \[ED=AC-AB \iff R=2r+r_a.\]

The inscribed circle of the triangle $ABC$ touches the sides $BC, CA, AB$ at points $A_1, B_1, C_1$ respectively. The points $P_b, Q_b, R_b$ are the points of the segments $BC_1, C_1A_1, A_1B$, respectively, such that $BP_bQ_bR_b$ is parallelogram. In the same way, the points $P_c, Q_c, R_c$ are the points of the sections $CB_1, B_1A_1, A_1C$, respectively such that $CP_cQ_cR_c$ is a parallelogram. The intersection of the lines $P_bR_b$ and $P_cR_c$ is $T$. Show that $TQ_b = TQ_c$.

Trapezoid $ ABCD$ has $ AD\parallel{}BC$, $ BD \equal{} 1$, $ \angle DBA \equal{} 23^{\circ}$, and $ \angle BDC \equal{} 46^{\circ}$. The ratio $ BC: AD$ is $ 9: 5$. What is $ CD$? $ \textbf{(A)}\ \frac {7}{9}\qquad \textbf{(B)}\ \frac {4}{5}\qquad \textbf{(C)}\ \frac {13}{15} \qquad \textbf{(D)}\ \frac {8}{9}\qquad \textbf{(E)}\ \frac {14}{15}$

Let be given a parallelogram $ ABCD$ and two points $ A_1$, $ C_1$ on its sides $ AB$, $ BC$, respectively. Lines $ AC_1$ and $ CA_1$ meet at $ P$. Assume that the circumcircles of triangles $ AA_1P$ and $ CC_1P$ intersect at the second point $ Q$ inside triangle $ ACD$. Prove that $ \angle PDA \equal{} \angle QBA$.

Let $ABC$ be an equilateral triangle with the side of $20$ units. Amir divides this triangle into $400$ smaller equilateral triangles with the sides of $1$ unit. Reza then picks $4$ of the vertices of these smaller triangles. The vertices lie inside the triangle $ABC$ and form a parallelogram with sides parallel to the sides of the triangle $ABC.$ There are exactly $46$ smaller triangles that have at least one point in common with the sides of this parallelogram. Find all possible values for the area of this parallelogram. [asy] unitsize(150); defaultpen(linewidth(0.7)); int n = 20; /* # of vertical lines, including BC */ pair A = (0,0), B = dir(-30), C = dir(30); draw(A--B--C--cycle,linewidth(1)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0)); label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE); for(int i = 1; i < n; ++i) { draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n); draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n); draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n); }[/asy] [Thanks azjps for drawing the diagram.] [hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]

Inside the parallelogram $ABCD$, point $M$ is chosen, and inside the triangle $AMD$, point $N$ is chosen in such a way that $$\angle MNA + \angle MCB =\angle MND + \angle MBC = 180^o.$$ Prove that lines $MN$ and $AB$ are parallel.

Two parallelograms are arranged so as it shown on the picture. Prove that the diagonal of the one parallelogram passes through the intersection point of the diagonals of the second. [img]https://cdn.artofproblemsolving.com/attachments/9/a/15c2f33ee70eec1bcc44f94ec0e809c9e837ff.png[/img]

Given triangle $ABC$. Point $B_1$ is marked on line $AC$ so that $AB = AB_1$, while $B_1$ and $C$ are on the same side of $A$. Through points $C$, $B_1$ and the foot of the bisector of angle $A$ of triangle $ABC$, a circle $\omega$ is drawn, intersecting for second time the circle circumscribed around triangle $ABC$, at point $Q$. Prove that the tangent drawn to $\omega$ at point $Q$ is parallel to $AC$.

Suppose $ABCD$ is a parallelogram and $E$ is a point between $B$ and $C$ on the line $BC$. If the triangles $DEC$, $BED$ and $BAD$ are isosceles what are the possible values for the angle $DAB$?

$ AA_{3}$ and $ BB_{3}$ are altitudes of acute-angled $ \triangle ABC$. Points $ A_{1}$ and $ B_{1}$ are second points of intersection lines $ AA_{3}$ and $ BB_{3}$ with circumcircle of $ \triangle ABC$ respectively. $ A_{2}$ and $ B_{2}$ are points on $ BC$ and $ AC$ respectively. $ A_{1}A_{2}\parallel AC$, $ B_{1}B_{2}\parallel BC$. Point $ M$ is midpoint of $ A_{2}B_{2}$. $ \angle BCA \equal{} x$. Find $ \angle A_{3}MB_{3}$.

Point $P$ lies inside of parallelogram $ABCD$. Perpendicular lines to $PA,PB,PC$ and $PD$ through $A,B,C$ and $D$ construct convex quadrilateral $XYZT$. Prove that $S_{XYZT}\geq 2S_{ABCD}$. [i]Proposed by Siamak Ahmadpour[/i]

Prove that if a lattice parallellogram contains an odd number of lattice points, then its centroid.

Let $O$ be the center of insphere of a tetrahedron $ABCD$. The sum of areas of faces $ABC$ and $ABD$ equals the sum of areas of faces $CDA$ and $CDB$. Prove that $O$ and midpoints of $BC, AD, AC$ and $BD$ belong to the same plane.

Given a parallelogram $ABCD$. A circle passing through $A$ meets the line segments $AB, AC$ and $AD$ at inner points $M,K,N$, respectively. Prove that \[|AB|\cdot |AM | + |AD|\cdot |AN|=|AK|\cdot |AC|\]

Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$

Let $ ABCD$ be a cyclic quadrilater. Denote $ P\equal{}AD\cap BC$ and $ Q\equal{}AB \cap CD$. Let $ E$ be the fourth vertex of the parallelogram $ ABCE$ and $ F\equal{}CE\cap PQ$. Prove that $ D,E,F$ and $ Q$ lie on the same circle.

Let $ABCD$ be a parallelogram. Let $M$ be the midpoint of the arc $AC$ containing $B$ of the circumcircle of $ABC$ . Let $E$ be a point on segment $AD$ and $F$ a point on segment $CD$ such that $ME=MD=MF$. Show that $BMEF$ is cyclic. [i]Proposed by A. Tereshin[/i]

The incircle of $ABC$ touches $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. The line through $A$ parallel to $LK$ meets $MK$ at $P$, and the line through $A$ parallel to $MK$ meets $LK$ at $Q$. Show that the line $PQ$ bisects $AB$ and bisects $AC$.

Given $\vartriangle ABC$ and a point $P$ inside that triangle. The parallelograms $CPBL$, $APCM$ and $BPAN$ are constructed. Prove that $AL$, $BM$ and $CN$ pass through one point $S$, and that $S$ is the midpoint of $AL$, $BM$ and $CN$.

a) Given a triangle $A_1A_2A_3$ and the points $B_1$ and $D_2$ on the side $[A_1A_2]$, $B_2$ and $D_3$ on the side $[A_2A3]$, $B_3$ and $D_1$ on the side $[A_3A_1]$. If you construct parallelograms $A_1B_1C_1D_1$, $A_2B_2C_2D_2$ and $A_3B_3C_3D_3$, the lines $(A_1C_1)$, $(A_2C_2)$ and $(A_3C_3)$, will cross in one point $O$. Prove that if $$|A_1B_1| = |A_2D_2| \,\,\, and \,\,\, |A_2B_2| = |A_3D_3|$$ then $$|A_3B_3| = |A_1D_1|$$ b) Given a convex polygon $A_1A_2 ... A_n$ and the points $B_1$ and $D_2$ on the side $[A_1A_2]$, $B_2$ and $D_3$ on the side $[A_2A_3]$, ... $B_n$ and $D_1$ on the side $[A_nA_1]$. If you construct parallelograms $A_1B_1C_1D_1$, $A_2B_2C_2D_2$, $... $, $A_nB_nC_nD_n$, the lines $(A_1C_1)$, $(A_2C_2)$, $...$, $(A_nC_n)$, will cross in one point $O$. Prove that $$|A_1B_1| \cdot |A_2B_2|\cdot ... \cdot |A_nB_n| = |A_1D_1|\cdot |A_2D_2|\cdot ...\cdot |A_nD_n|$$

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.