Found problems: 1049
Let $ABC$ be a triangle . Let point $X$ be in the triangle and $AX$ intersects $BC$ in $Y$ . Draw the perpendiculars $YP,YQ,YR,YS$ to lines $CA,CX,BX,BA$ respectively. Find the necessary and sufficient condition for $X$ such that $PQRS$ be cyclic .
A quadrilateral $ABCD$ is inscribed in a circle $\omega$. The tangent to $\omega$ at $A$ intersects the ray $CB$ at $K$, and the tangent to $\omega$ at $B$ intersects the ray $DA$ at $M$. Prove that if $AM=AD$ and $BK=BC$, then $ABCD$ is a trapezoid.
On each side of a parallelogram an arbitrary point is chosen. Each pair of chosen points on neighbouring sides (i.e. sides with a common vertex) are connected by a line segment. Prove that the centres of the circumscribed circles of the four triangles so created are themselves vertices of a parallelogram.
(ED Kulanin)
Suppose $ABCD$ is a parallelogram. Consider circles $w_1$ and $w_2$ such that $w_1$ is tangent to segments $AB$ and $AD$ and $w_2$ is tangent to segments $BC$ and $CD$. Suppose that there exists a circle which is tangent to lines $AD$ and $DC$ and externally tangent to $w_1$ and $w_2$. Prove that there exists a circle which is tangent to lines $AB$ and $BC$ and also externally tangent to circles $w_1$ and $w_2$.
[i]Proposed by Ali Khezeli[/i]
Circles $\alpha$ and $\beta$ of the same radius intersect in two points, one of which is $P$. Denote by $A$ and $B$, respectively, the points diametrically opposite to $P$ on each of $\alpha$ and $\beta$ . A third circle of the same radius passes through $P$ and intersects $\alpha$ and $\beta$ in the points $X$ and $Y$ , respectively. Show that the line $XY$ is parallel to the line $AB$.
Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that:
\[PX || AC \ , \ PY ||AB \]
Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$
Let $ABC$ be a triangle and $D$, $E$, $F$ be the midpoints of arcs $BC$, $CA$, $AB$ on the circumcircle. Line $\ell_a$ passes through the feet of the perpendiculars from $A$ to $DB$ and $DC$. Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$. Let $A_1$ denote the intersection of lines $\ell_a$ and $m_a$. Define points $B_1$ and $C_1$ similarly. Prove that triangle $DEF$ and $A_1B_1C_1$ are similar to each other.
Let $A, B$ and $C$ be three non-collinear points and $E$ ($\ne B$) an arbitrary point not in the straight line $AC$. Construct the parallelograms $ABCD$ and $AECF$. Prove that $BE \parallel DF$.
Let be given a triangle $ ABC$. Points $ P$ on side $ AC$ and $ Y$ on the production of $ CB$ beyond $ B$ are chosen so that $ Y$ subtends equal angles with $ AP$ and $ PC$. Similarly, $ Q$ on side $ BC$ and $ X$ on the production of $ AC$ beyond $ C$ are such that $ X$ subtends equal angles with $ BQ$ and $ QC$. Lines $ YP$ and $ XB$ meet at $ R$, $ XQ$ and $ YA$ meet at $ S$, and $ XB$ and $ YA$ meet at $ D$. Prove that $ PQRS$ is a parallelogram if and only if $ ACBD$ is a cyclic quadrilateral.
Let $O$ be the center of the circumcircle $\omega$ of an acute-angle triangle $ABC$. A circle $\omega_1$ with center $K$ passes through $A$, $O$, $C$ and intersects $AB$ at $M$ and $BC$ at $N$. Point $L$ is symmetric to $K$ with respect to line $NM$. Prove that $BL \perp AC$.
In a parallelogram $ABCD$, let $M$ be the point on the $BC$ side such that $MC = 2BM$ and let $N$ be the point of side $CD$ such that $NC = 2DN$. If the distance from point $B$ to the line $AM$ is $3$, calculate the distance from point $N$ to the line $AM$.
Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$, respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$, respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$, and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$. Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$. Prove that $AQ=D_2P$.
The circles $ c_1$ and $ c_2$ are tangent at the point $ A.$ A straight line $ l$ through $ A$ intersects $ c_1$ and $ c_2$ at points $ C_1$ and $ C_2$ respectively. A circle $ c,$ which contains $ C_1$ and $ C_2,$ meets $ c_1$ and $ c_2$ at points $ B_1$ and $ B_2$ respectively. Let $ \omega$ be the circle circumscribed around triangle $ AB_1B_2.$ The circle $ k$ tangent to $ \omega$ at the point $ A$ meets $ c_1$ and $ c_2$ at the points $ D_1$ and $ D_2$ respectively. Prove that
[b](a)[/b] the points $ C_1,C_2,D_1,D_2$ are concyclic or collinear,
[b](b)[/b] the points $ B_1,B_2,D_1,D_2$ are concyclic if and only if $ AC_1$ and $ AC_2$ are diameters of $ c_1$ and $ c_2.$
The diagonals of parallelogram $ABCD$ intersect at $E$ . The bisectors of $\angle DAE$ and $\angle EBC$ intersect at $F$. Assume $ECFD$ is a parellelogram . Determine the ratio $AB:AD$.
Let $ABC$ be an acute angles triangle with $O$ the center of the circumscribed circle. Point $Q$ lies on the circumscribed circle of $\vartriangle BOC$ so that $OQ$ is a diameter. Point $M$ lies on $CQ$ and point $N$ lies internally on line segment $BC$ so that $ANCM$ is a parallelogram. Prove that the circumscribed circle of $\vartriangle BOC$ and the lines $AQ$ and $NM$ pass through the same point.
$ABCD$ is a quadrilateral such that $AB$ is not parallel to $CD$, and $BC$ is not parallel to $AD$. Variable points $P, Q, R, S$ are taken on $AB, BC, CD, DA$ respectively so that $PQRS$ is a parallelogram. Find the locus of its center.
In isosceles triangle $ABC$ ($AB = BC$) one draws the angle bisector $CD$. The perpendicular to $CD$ through the center of the circumcircle of $ABC$ intersects $BC$ at $E$. The parallel to $CD$ through $E$ meets $AB$ at $F$. Show that $BE$ = $FD$.
[i]M. Sonkin[/i]
Let $ABC$ be an acute triangle inscribed in a circle of center $O$. If the altitudes $BD,CE$ intersect at $H$ and the circumcenter of $\triangle BHC$ is $O_1$, prove that $AHO_1O$ is a parallelogram.
Given is a parallelogram $XYZT$, and the variable points $A, B, C, D$ lie on the sides $XY, XT, TZ, ZY$ respectively, so that $ABCD$ is cyclic with circumcenter $O$, $AC \parallel XT$, and $BD \parallel XY$. Let $P$ be the intersection point of the lines $AD$ and $BC$, and let $Q$ be the intersection of the lines $AB$ and $CD$. Prove that the circle $(POQ)$ passes through a fixed point.
Let $ABCD$ be a parallelogram with at least an angle not equal to $90^o$ and $k$ the circumcircle of the triangle $ABC$. Let $E$ be the diametrically opposite point of $B$. Show that the circumcircle of the triangle $ADE$ and $k$ have the same radius.
Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
In the parallelogram $ABCD$, $\angle BAD =60 ^ \circ$. Let $E $ be the intersection point of the diagonals. The circle circumscribed to the triangle $ACD$ intersects the line $AB$ at the point $K$ (different from $A$), the line $BD$ at the point $P$ (different from $D$), and to the line $BC$ in $L$ (different from $C$). The line $EP$ intersects the circumscribed circle of the triangle $CEL$ at the points $E$ and $M$. Show that the triangles $KLM$ and $CAP$ are congruent.
A circle $K$ with radius $r$, a point $D$ on $K$, and a convex angle with vertex $S$ and rays $a$ and $b$ are given in the plane. Construct a parallelogram $ABCD$ such that $A$ and $B$ lie on $a$ and $b$ respectively, $SA+SB=r$, and $C$ lies on $K$.
Let the medians of the triangle $ABC$ meet at $G$. Let $D$ and $E$ be different points on the line $BC$ such that $DC=CE=AB$, and let $P$ and $Q$ be points on the segments $BD$ and $BE$, respectively, such that $2BP=PD$ and $2BQ=QE$. Determine $\angle PGQ$.
Given triangle $ ABC$ with medians $ AE$, $ BF$, $ CD$; $ FH$ parallel and equal to $ AE$; $ BH$ and $ HE$ are drawn; $ FE$ extended meets $ BH$ in $ G$. Which one of the following statements is not necessarily correct?
$ \textbf{(A)}\ AEHF \text{ is a parallelogram} \qquad
\textbf{(B)}\ HE\equal{}HG \\
\textbf{(C)}\ BH\equal{}DC \qquad
\textbf{(D)}\ FG\equal{}\frac{3}{4}AB \qquad
\textbf{(E)}\ FG\text{ is a median of triangle }BFH$