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 $s=\frac{AB+BC+AC}{2}$ be half-perimeter of triangle $ABC$. Let $L$ and $N$be a point's on ray's $AB$ and $CB$, for which $AL=CN=s$. Let $K$ is point, symmetric of point $B$ by circumcenter of $ABC$. Prove, that perpendicular from $K$ to $NL$ passes through incenter of $ABC$. Solution for problem [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

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]

Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.

Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure). Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square. [asy] unitsize(1 cm); pair A, B, C, D, E, F, O, X, Y, Z; X = (1,4); Y = (0,0); Z = (5,1.5); O = (1.8,2.2); A = extension(O, O + Z - X, X, Y); B = extension(O, O + Y - Z, X, Y); C = extension(O, O + X - Y, Y, Z); D = extension(O, O + Z - X, Y, Z); E = extension(O, O + Y - Z, Z, X); F = extension(O, O + X - Y, Z, X); draw(X--Y--Z--cycle); draw(A--D); draw(B--E); draw(C--F); dot("$A$", A, NW); dot("$B$", B, NW); dot("$C$", C, SE); dot("$D$", D, SE); dot("$E$", E, NE); dot("$F$", F, NE); dot("$O$", O, S); dot("$X$", X, N); dot("$Y$", Y, SW); dot("$Z$", Z, dir(0)); label("$a$", (A + O)/2, SW); label("$b$", (B + O)/2, SE); label("$c$", (C + O)/2, SE); label("$d$", (D + O)/2, SW); label("$e$", (E + O)/2, SE); label("$f$", (F + O)/2, NW); [/asy]

In the parallelogram $ABCD$, the diagonal $AC$ touches the incircles of triangles $ABC$ and $ADC$ at $W$ and $Y$ respectively, and the diagonal $BD$ touches the incircles of triangles $BAD$ and $BCD$ at $X$ and $Z$ respectively. Prove that either $W,X, Y$ and $Z$ coincide, or $WXYZ$ is a rectangle.

Let $E$ be the point of intersection of the diagonals of convex quadrilateral $ABCD$, and let $P,Q,R,$ and $S$ be the centers of the circles circumscribing triangles $ABE,$ $BCE$, $CDE$, and $ADE$, respectively. Then $\textbf{(A) }PQRS\text{ is a parallelogram}$ $\textbf{(B) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a rhombus}$ $\textbf{(C) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a rectangle}$ $\textbf{(D) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a parallelogram}$ $\textbf{(E) }\text{none of the above are true}$

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.

Let $I$ be the incentre of triangle $ABC$. A circle containing the points $B$ and $C$ meets the segments $BI$ and $CI$ at points $P$ and $Q$ respectively. It is known that $BP\cdot CQ=PI\cdot QI$. Prove that the circumcircle of the triangle $PQI$ is tangent to the circumcircle of $ABC$. [i]Proposed by S. Berlov[/i]

A circle touches sides $AB, BC, CD$ of a parallelogram $ABCD$ at points $K, L, M$ respectively. Prove that the line $KL$ bisects the height of the parallelogram drawn from the vertex $C$ to $AB$.

Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.

Let $ABCD$ be a cyclic quadrilateral in which internal angle bisectors $\angle ABC$ and $\angle ADC$ intersect on diagonal $AC$. Let $M$ be the midpoint of $AC$. Line parallel to $BC$ which passes through $D$ cuts $BM$ at $E$ and circle $ABCD$ in $F$ ($F \neq D$ ). Prove that $BCEF$ is parallelogram [i]Proposed by Steve Dinh[/i]

Let $P$ and $Q$ be the points on the diagonal $BD$ of a quadrilateral $ABCD$ such that $BP = PQ = QD$. Let $AP$ and $BC$ meet at $E$, and let $AQ$ meet $DC$ at $F$. (a) Prove that if $ABCD$ is a parallelogram, then $E$ and $F$ are the midpoints of the corresponding sides. (b) Prove the converse of (a).

In parallelogram $ABCD$, point $K$ is marked on diagonal $AC$. Circle $s_1$ passes through point $K$ and touches lines $AB$ and $AD$ ($s_1$ intersects the diagonal $AC$ for the second time on the segment $AK$). Circle $s_2$ passes through point $K$ and touches lines $CB$ and $CD$ ($s_2$ intersects for the second time diagonal $AC$ on segment $KC$). Prove that for all positions of the point $K$ on the diagonal $AC$, the straight lines connecting the centers of circles $ s_1$ and $s_2$, will be parallel to each other.

The points $P$ and $Q$ lie on the sides $BC$ and $CD$ of the parallelogram $ABCD$ so that $BP = QD$. Show that the intersection point between the lines $BQ$ and $DP$ lies on the line bisecting $\angle BAD$.

Let $A_{1}A_{2}A_{3}$ be an acute triangle, and let $O$ and $H$ be its circumcenter and orthocenter, respectively. For $1\leq i \leq 3$, points $P_{i}$ and $Q_{i}$ lie on lines $OA_{i}$ and $A_{i+1}A_{i+2}$ (where $A_{i+3}=A_{i}$), respectively, such that $OP_{i}HQ_{i}$ is a parallelogram. Prove that \[\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.\]

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 $\angle AED+\angle BEC=180^\circ$.

Geodude wants to assign one of the integers $1,2,3,\ldots,11$ to each lattice point $(x,y,z)$ in a 3D Cartesian coordinate system. In how many ways can Geodude do this if for every lattice parallelogram $ABCD$, the positive difference between the sum of the numbers assigned to $A$ and $C$ and the sum of the numbers assigned to $B$ and $D$ must be a multiple of $11$? (A [i]lattice point[/i] is a point with all integer coordinates. A [i]lattice parallelogram[/i] is a parallelogram with all four vertices lying on lattice points. Here, we say four not necessarily distinct points $A,B,C,D$ form a [i]parallelogram[/i] $ABCD$ if and only if the midpoint of segment $AC$ coincides with the midpoint of segment $BD$.) [hide="Clarifications"] [list] [*] The ``positive difference'' between two real numbers $x$ and $y$ is the quantity $|x-y|$. Note that this may be zero. [*] The last sentence was added to remove confusion about ``degenerate parallelograms.''[/list][/hide] [i]Victor Wang[/i]

In the acute-angled triangle $ ABC $, the sides $ AB $ and $BC$ have different lengths, and the extension of the median $ BM $ intersects the circumscribed circle at the point $ N $. On this circle we note such a point $ D $ that $ \angle BDH = 90 {} ^ \circ $, where $ H $ is the point of intersection of the altitudes of the triangle $ ABC $. The point $K$ is chosen so that $ ANCK $ is a parallelogram. Prove that the lines $ AC $, $ KH $ and $ BD $ intersect at one point. (Igor Nagel)

Given a circle $K$, find the locus of vertices $A$ of parallelograms $ABCD$ with diagonals $AC \leq BD$, such that $BD$ is inside $K$.

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 $ABCD$ be a parallelogram. Extend $\overline{DA}$ through $A$ to a point $P,$ and let $\overline{PC}$ meet $\overline{AB}$ at $Q$ and $\overline{DB}$ at $R.$ Given that $PQ=735$ and $QR=112,$ find $RC.$

In a triangle $ABC$, let $H$ be the point of intersection of altitudes, $I$ the center of incircle, $O$ the center of circumcircle, $K$ the point where the incircle touches $BC$. Given that $IO$ is parallel to $BC$, prove that $AO$ is parallel to $HK$.

Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.

Let $ABCD$ be a parallelogram. Let $\omega_1$ be the circle passing through $D$ tangent to $AB$ at $A$. Let $\omega_2$ be the circle passing through $A$ tangent to $CD$ at $D$. The tangents from $B$ to $\omega_1$ touch it at $A$ and $P$. The tangents from $C$ to $\omega_2$ touch it at $D$ and $Q$. Lines $AP$ and $DQ$ intersect at $X$. The perpendicular bisector of $BC$ intersects $AD$ at $R$. Show that the circumcircles of triangles $\triangle PQX$, $\triangle BCR$ are concentric.