Found problems: 1049
Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
Quadrilateral $ABCD$ is a parallelogram, and $E$ is the midpoint of the side $\overline{AD}$. Let $F$ be the intersection of lines $EB$ and $AC$. What is the ratio of the area of quadrilateral $CDEF$ to the area of triangle $CFB$?
$\textbf{(A) } 5 : 4 \qquad \textbf{(B) } 4 : 3 \qquad \textbf{(C) } 3 : 2 \qquad \textbf{(D) } 5 : 3 \qquad \textbf{(E) } 2 : 1$
Let $E$ and $F$ are distinct points on the diagonal $AC$ of a parallelogram $ABCD$ . Prove that , if there exists a cricle through $E,F$ tangent to rays $BA,BC$ then there also exists a cricle through $E,F$ tangent to rays $DA,DC$.
Triangle $ABC$ is given. Let $M$ be the midpoint of the segment $AB$ and $T$ be the midpoint of the arc $BC$ not containing $A$ of the circumcircle of $ABC.$ The point $K$ inside the triangle $ABC$ is such that $MATK$ is an isosceles trapezoid with $AT\parallel MK.$ Show that $AK = KC.$
In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$
such that $BC$ not is the diameter.Consider a point $A$ varies in
$(O)$ such that $A\neq B,C$ and $AB\neq AC$.Call $D$ and $E$
respective is intersect of $BC$ and internal and external bisector
of $\widehat{BAC}$,$I$ is midpoint of $DE$.The line that pass through
orthocenter of $\triangle ABC$
and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$.
1/Prove that $MN$ pass through a fixed point
2/Determint the place of $A$ such that $S_{AMN}$ has maxium value
The ratio of the sides of a parallelogram is $\lambda>1$. Given $\lambda$, determine the maximum of the acute angle subtended by the diagonals of the parallelogram.
Let $ABC$ be an acute-angled triangle, $H$ the intersection point of its altitudes , and $AA'$ the diameter of the circumcircle of triangle $ABC$. Prove that the quadrilateral $HB A'C$ is a parallelogram.
The points of intersection of $ xy \equal{} 12$ and $ x^2 \plus{} y^2 \equal{} 25$ are joined in succession. The resulting figure is:
$ \textbf{(A)}\ \text{a straight line} \qquad\textbf{(B)}\ \text{an equilateral triangle} \qquad\textbf{(C)}\ \text{a parallelogram}$
$ \textbf{(D)}\ \text{a rectangle} \qquad\textbf{(E)}\ \text{a square}$
$BC$ is a diameter of a circle and the points $X,Y$ are on the circle such that $XY\perp BC$. The points $P,M$ are on $XY,CY$ (or their stretches), respectively, such that $CY||PB$ and $CX||PM$. Let $K$ be the meet point of the lines $XC,BP$. Prove that $PB\perp MK$.
In a triangle $\triangle ABC$ points $L, P$ and $Q$ lie on the segments $AB, AC$ and $BC$, respectively, and are such that $PCQL$ is a parallelogram. The circle with center the midpoint $M$ of the segment $AB$ and radius $CM$ and the circle of diameter $CL$ intersect for the second time at the point $T$. Prove that the lines $AQ, BP$ and $LT$ intersect in a point.
Prove that if a lattice parallellogram contains an odd number of lattice points, then its centroid.
Let $ABC$ be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to $A$.
Suppose that the outer inscribed circle with respect to $A$ touches $BC,AC,AB$ at $M,N,L$.
Prove that $O$ (Center of circumcircle) is the orthocenter of $MNL$.
Let $ABCD$ be a quadrilateral (with non-perpendicular diagonals).
The perpendicular from $A$ to $BC$ meets $CD$ at $K$.
The perpendicular from $A$ to $CD$ meets $BC$ at $L$.
The perpendicular from $C$ to $AB$ meets $AD$ at $M$.
The perpendicular from $C$ to $AD$ meets $AB$ at $N$.
1. Prove that $KL$ is parallel to $MN$.
2. Prove that $KLMN$ is a parallelogram if $ABCD$ is cyclic.
In a triangle $A_1A_2A_3$, the excribed circles corresponding to sides $A_2A_3$, $A_3A_1$, $A_1A_2$ touch these sides at $T_1$, $T_2$, $T_3$, respectively. If $H_1$, $H_2$, $H_3$ are the orthocenters of triangles $A_1T_2T_3$, $A_2T_3T_1$, $A_3T_1T_2$, respectively, prove that lines $H_1T_1$, $H_2T_2$, $H_3T_3$ are concurrent.
Three equal circles $k_1, k_2, k_3$ intersect non-tangentially at a point $P$. Let $A$ and $B$ be the centers of circles $k_1$ and $k_2$. Let $D$ and $C$ be the intersection of $k_3$ with $k_1$ and $k_2$ respectively, which is different from $P$. Show that $ABCD$ is a parallelogram.
The parallelogram bounded by the lines $ y \equal{} ax \plus{} c,y \equal{} ax \plus{} d,y \equal{} bx \plus{} c$ and $ y \equal{} bx \plus{} d$ has area $ 18$. The parallelogram bounded by the lines $ y \equal{} ax \plus{} c,y \equal{} ax \minus{} d,y \equal{} bx \plus{} c,$ and $ y \equal{} bx \minus{} d$ has area $ 72.$ Given that $ a,b,c,$ and $ d$ are positive integers, what is the smallest possible value of $ a \plus{} b \plus{} c \plus{} d$?
$ \textbf{(A)}\ 13 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17$
Compute the maximum area of a rectangle which can be inscribed in a triangle of area $M$.
Given a parallelogram $ABCD$ with $AB<BC$, show that the circumcircles of the triangles $APQ$ share a second common point (apart from $A$) as $P,Q$ move on the sides $BC,CD$ respectively s.t. $CP=CQ$.
Let $ABCD$ be a parallelogram. On side $AB$, point $M$ is taken so that $AD = DM$. On side $AD$ point $N$ is taken so that $AB = BN$. Prove that $CM = CN$.
Let $O, H$ be the circumcentor and the orthocenter of a scalene triangle $ABC$. Let $P$ be the reflection of $A$ w.r.t. $OH$, and $Q$ is a point on $\odot (ABC)$ such that $AQ, OH, BC$ are concurrent. Let $A'$ be a points such that $ABA'C$ is a parallelogram. Show that $A', H, P, Q$ are concylic.
(ltf0501).
Complex number $z=\cos\theta+\text{i}\sin\theta(0\leq\theta\leq\pi)$. Points that three complex numbers $z,(1+\text{i})z,2\overline{z}$ refer to on complex plane are $P,Q,R$. When $P,Q,R$ are not collinear, $PQSR$ is a parallelogram. The longest distance between $S$ and the original point is________.
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 $ABCD$ be a parallelogram whose diagonals meet at $P$. Denote by $M$ the midpoint of $AB$. Let $Q$ be a point such that $QA$ is tangent to the circumcircle of $MAD$ and $QB$ is tangent to the circumcircle of $MBC$. Prove that points $Q,M,P$ are collinear.
(Patrik Bak, Slovakia)
Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
A rectangle is said to be inscribed in a parallelogram if its vertices lie one on each side of the parallelogram. On the larger side $AB$ of the parallelogram $ABCD$, find all those points $K$ that are the vertices of the rectangles inscribed in $ABCD$.