Let $ABCD$ be a parallelogram and $P$ be an arbitrary point of the circumcircle of $\Delta ABD$, different from the vertices. Line $PA$ intersects the line $CD$ at point $Q$. Let $O$ be the center of the circumcircle $\Delta PCQ$. Prove that $\angle ADO = 90^o$.

Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 32$ and $GF = 24$, then $BE$ equals: [asy] size(7cm); pair A = (0, 0), B = (7, 0), C = (10, 5), D = (3, 5), F = (5.7, 9.5); pair G = intersectionpoints(B--F, D--C)[0]; pair E = intersectionpoints(A--C, B--F)[0]; draw(A--D--C--B--cycle); draw(A--C); draw(D--F--B); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, NE); label("$D$", D, NW); label("$F$", F, N); label("$G$", G, NE); label("$E$", E, SE); //Credit to MSTang for the asymptote [/asy] $\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 16$

Let $[ABCD]$ be a parallelogram and $P$ a point between $C$ and $D$. The line parallel to $AD$ that passes through $P$ intersects the diagonal $AC$ in $Q$. Knowing that the area of $[PBQ]$ is $2$ and the area of $[ABP]$ is $6$, determine the area of $[PBC]$. [img]https://cdn.artofproblemsolving.com/attachments/0/8/664a00020065b7ad6300a062613fca4650b8d0.png[/img]

In a given plane $\varrho$ consider a convex quadrilateral $ABCD$ and denote $E=AC\cap BD.$ Moreover, consider a point $V\notin\varrho$. On rays $VA,VB,VC,VD$ find points $A',B',C',D'$ respectively such that $E,A',B',C',D'$ are coplanar and $A'B'C'D'$ is a parallelogram. Discuss conditions of solvability.

Let $ABCD$ be a cyclic quadrilateral such that the triangles $BCD$ and $CDA$ are not equilateral. Prove that if the Simson line of $A$ with respect to $\triangle BCD$ is perpendicular to the Euler line of $BCD$, then the Simson line of $B$ with respect to $\triangle ACD$ is perpendicular to the Euler line of $\triangle ACD$.

A point $K$ is taken inside parallelogram $ABCD$ so that the midpoint of $AD$ is equidistant from $K$ and $C$, and the midpoint of $CD$ is equidistant form $K$ and $A$. Let $N$ be the midpoint of $BK$. Prove that the angles $NAK$ and $NCK$ are equal.

Parallelogram $ABCD$ is such that angle $B < 90$ and $AB<BC$. Points E and F are on the circumference of $\omega$ inscribing triangle ABC, such that tangents to $\omega$ in those points pass through D. If $\angle EDA= \angle{FDC}$, find $\angle{ABC}$.

Points $E$ and $F$ lying on sides $BC$ and $AD$ respectively of a parallelogram $ABCD$ are such that $EF=ED=DC$. Let $M$ be the midpoint of $BE$ and $MD$ meet $EF$ at $G$. Prove that $\angle EAC=\angle GBD$.

In parallelogram $ABCD$, points $E$ and $F$ on segments $AD$ and $CD$ are such that $\angle BCE=\angle BAF$. Points $K$ and $L$ on segments $AD$ and $CD$ are such that $AK=ED$ and $CL=FD$. Prove that $\angle BKD=\angle BLD$.

A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs. Juniors) How many vertical sides can there be in this set? Seniors) How many ways are there to do that? [asy] size(120); defaultpen(linewidth(0.8)); path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle; for(int i=0;i<=3;i=i+1) { for(int j=0;j<=2;j=j+1) { real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2; draw(shift(shiftx,shifty)*hex); } } [/asy] [i](T. Doslic)[/i]

Squares $ BCA_{1}A_{2}$ , $ CAB_{1}B_{2}$ , $ ABC_{1}C_{2}$ are outwardly drawn on sides of triangle $ \triangle ABC$. If $ AB_{1}A'C_{2}$ , $ BC_{1}B'A_{2}$ , $ CA_{1}C'B_{2}$ are parallelograms then prove that: (i) Lines $ BC$ and $ AA'$ are orthogonal. (ii)Triangles $ \triangle ABC$ and $ \triangle A'B'C'$ have common centroid

Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$.

Let $\omega_P$ and $\omega_Q$ be two circles of radius $1$, intersecting in points $A$ and $B$. Let $P$ and $Q$ be two regular $n$-gons (for some positive integer $n\ge4$) inscribed in $\omega_P$ and $\omega_Q$, respectively, such that $A$ and $B$ are vertices of both $P$ and $Q$. Suppose a third circle $\omega$ of radius $1$ intersects $P$ at two of its vertices $C$, $D$ and intersects $Q$ at two of its vertices $E$, $F$. Further assume that $A$, $B$, $C$, $D$, $E$, $F$ are all distinct points, that $A$ lies outside of $\omega$, and that $B$ lies inside $\omega$. Show that there exists a regular $2n$-gon that contains $C$, $D$, $E$, $F$ as four of its vertices.

Let $ABC$ be an isosceles triangle with $AC=BC$. Let $P,Q,R$ be points on the sides $AB, BC, CA$ of the triangle such that $CQPR$ is a parallelogram. Show that the reflection of $P$ over $QR$ lies on the circumcircle of $ABC$.

Consider a parallelogram $ABCD$. a) Prove that if the incenter of the triangle $ABC$ is located on the diagonal $BD$, then the parallelogram $ABCD$ is a rhombus. b) Is the parallelogram $ABCD$ a rhombus whenever the circumcenter of the triangle $ABC$ is located on the diagonal $BD$?

The sides of the parallelogram serve as the diagonals of the four squares. The vertices of the squares lying in the part of the plane external to the parallelogram (the sides of the squares emerging from these vertices do not have common points with the parallelogram) serve as the vertices of a quadrilateral of area $a$, the four vertices opposite to them form a quadrilateral of area $b$. Find the area of the parallelogram.

Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.

$AB$ is the diameter of semicircle $O$. $C$,$D$ are two arbitrary points on semicircle $O$. Point $P$ lies on line $CD$ such that line $PB$ is tangent to semicircle $O$ at $B$. Line $PO$ intersects line $CA$, $AD$ at point $E$, $F$ respectively. Prove that $OE$=$OF$.

Let $ABCD$ be a parallelogram and $\omega$ be the circumcircle of the triangle $ABD$. Let $E ,F$ be the intersections of $\omega$ with lines $BC ,CD$ respectively . Prove that the circumcenter of the triangle $CEF$ lies on $\omega$.

The vertex $ A$ of the acute triangle $ ABC$ is equidistant from the circumcenter $ O$ and the orthocenter $ H.$ Determine all possible values for the measure of angle $ A.$

Let $\vartriangle ABC$ be an isosceles triangle with $AC = BC$. Points $D, E, F$ are considered on $BC, CA, AB$, respectively, such that $AF> BF$ and that the quadrilateral $CEFD$ is a parallelogram. The perpendicular line to $BC$ drawn by $B$ intersects the perpendicular bisector of $AB$ at $G$. Prove that $DE \perp FG$.

Let $ABC$ be an acute scalene triangle, $H$ its orthocenter and $G$ its geocenter. The circumference with diameter $AH$ cuts the circumcircle of $BHC$ in $A'$ ($A' \neq H$). Points $B'$ and $C'$ are defined similarly. Show that the points $A'$, $B'$, $C'$, and $G$ lie in one circumference.

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. [i]Proposed by Sammy Luo[/i]

The sphere $ \omega $ passes through the vertex $S$ of the pyramid $SABC$ and intersects with the edges $SA,SB,SC$ at $A_1,B_1,C_1$ other than $S$. The sphere $ \Omega $ is the circumsphere of the pyramid $SABC$ and intersects with $ \omega $ circumferential, lies on a plane which parallel to the plane $(ABC)$. Points $A_2,B_2,C_2$ are symmetry points of the points $A_1,B_1,C_1$ respect to midpoints of the edges $SA,SB,SC$ respectively. Prove that the points $A$, $B$, $C$, $A_2$, $B_2$, and $C_2$ lie on a sphere.

Two sides of a quadrilateral $ABCD$ are parallel. Let $M$ and $N$ be the midpoints of $BC$ and $CD$ respectively, and $P$ be the intersection point of $AN$ and $DM$. Prove that if $AP=4PN$, then $ABCD$ is a parallelogram.