Found problems: 250
Let $k$ be the incircle of the triangle $ABC$ with the center of the incircle $I$. The circle $k$ touches the sides $BC, CA$ and $AB$ in points $D, E$ and $F$. Let $G$ be the intersection of the straight line $AI$ and the circle $k$, which lies between $A$ and $I$. Assume $BE$ and $FG$ are parallel. Show that $BD = EF$.
Let $ABCD$ be a convex quadrilateral with non perpendicular diagonals and with the sides $AB$ and $CD$ non parallel . Denote by $O$ the intersection of the diagonals , $H_1$ the orthocenter of the triangle $AOB$ and $H_2$ the orthocenter of the triangle $COD$ . Also denote with $M$ the midpoint of the side $AB$ and with $N$ the midpoint of the side $CD$ . Prove that $H_1H_2$ and $MN$ are parallel if and only if $AC=BD$
Six different points $A, B, C, D, E, F$ are marked on the plane, no four of them lie on one circle and no two segments with ends at these points lie on parallel lines. Let $P, Q,R$ be the points of intersection of the perpendicular bisectors to pairs of segments $(AD, BE)$, $(BE, CF)$ ,$(CF, DA)$ respectively, and $P', Q' ,R'$ are points the intersection of the perpendicular bisectors to the pairs of segments $(AE, BD)$, $(BF, CE)$ , $(CA, DF)$ respectively. Show that $P \ne P', Q \ne Q', R \ne R'$, and prove that the lines $PP', QQ'$ and $RR'$ intersect at one point or are parallel.
Let $O$ be the circumcentre of the acute triangle $ABC$ and let lines $AO$ and $BC$ intersect at point $K$. On sides $AB$ and $AC$, points $L$ and $M$ are chosen such that $|KL|= |KB|$ and $|KM| = |KC|$. Prove that segments $LM$ and $BC$ are parallel.
Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.
Let angle bisectors of angles $\angle BAC$ and $\angle ABC$ of triangle $ABC$ intersect sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let points $F$ and $G$ be foots of perpendiculars from point $C$ on lines $AD$ and $BE$, respectively. Prove that $FG \mid \mid AB$
Circles $k_1,k_2$ intersect at $B,C$ such that $BC$ is diameter of $k_1$.Tangent of $k_1$ at $C$ touches $k_2$ for the second time at $A$.Line $AB$ intersects $k_1$ at $E$ different from $B$, and line $CE$ intersects $k_2$ at F different from $C$. An arbitrary line through $E$ intersects segment $AF$ at $H$ and $k_1$ for the second time at $G$.If $BG$ and $AC$ intersect at $D$, prove $CH//DF$ .
In a triangle $ABC, O$ is the center of the circumscribed circle. Line $a$ passes through the midpoint of the altitude of the triangle from the vertex $A$ and is parallel to $OA$. Similarly, the straight lines $b$ and $c$ are defined. Prove that these three lines intersect at one point.
Given a plane with two circles, one with points $A$ and $B$, and the other with points $C$ and $D$ are shown in the figure. The line $AB$ passes through the center of the first circle and touches the second circle while the line $CD$ passes through the center of the second circle and touches the first circle. Prove that the lines $AD$ and $BC$ are parallel.
[img]https://cdn.artofproblemsolving.com/attachments/e/e/92f7b57751e7828a6487a052d4869e27e658b2.png[/img]
Let $ABC$ be an acute triangle with $AB < AC$. The points $A_1$ and $A_2$ are located on the line $BC$ so that $AA_1$ and $AA_2$ are the inner and outer angle bisectors at $A$ for the triangle $ABC$. Let $A_3$ be the mirror image $A_2$ with respect to $C$, and let $Q$ be a point on $AA_1$ such that $\angle A_1QA_3 = 90^o$. Show that $QC // AB$.
Given triangle $ABC$. Points $A_1,B_1$ and $C_1$ are symmetric to its vertices with respect to opposite sides. $C_2$ is the intersection point of lines $AB_1$ and $BA_1$. Points$ A_2$ and $B_2$ are defined similarly. Prove that the lines $A_1 A_2, B_1 B_2$ and $C_1 C_2$ are parallel.
(A. Zaslavsky)
Let $ABC$ a triangle with circumcircle $\Gamma$ and $R$ a point inside $ABC$ such that $\angle ABR=\angle RBC$. Let $\Gamma_1$ and $\Gamma_2$ the circumcircles of triangles $ARB$ and $CRB$ respectly. The parallel to $AC$ that pass through $R$, intersect $\Gamma$ in $D$ and $E$, with $D$ on the same side of $BR$ that $A$ and $E$ on the same side of $BR$ that $C$. $AD$ intersect $\Gamma_1$ in $P$ and $CE$ intersect $\Gamma_2$ in $Q$. Prove that $APQC$ is cyclic if and only if $AB=BC$
Let $ABC$ be a triangle, $I_a$ the center of the excircle at side $BC$, and $M$ its reflection across $BC$. Prove that $AM$ is parallel to the Euler line of the triangle $BCI_a$.
Let $\Omega$ be the circle circumscribed to the triangle $ABC$. Tangents taken to the circle $\Omega$ at points $A$ and $B$ intersects at the point $P$ , and the perpendicular bisector of $ (BC)$ cuts line $AC$ at point $Q$. Prove that lines $BC$ and $PQ$ are parallel.
Two straight lines $a$ and $b$ are given and also points $A$ and $B$. Point $X$ slides along the line $a$, and point $Y$ slides along the line $b$, so that $AX \parallel BY$. Find the locus of the intersection point of $AY$ with $XB$.
Given is the non-equilateral triangle $A_1A_2A_3$. $B_{ij}$ is the symmetric of $A_i$ wrt the inner bisector of $\angle A_j$. Prove that lines $B_{12}B_{21}$, $B_{13}B_{31}$ and $B_{23}B_{32}$ are parallel.
Let $ABCD$ be a convex quadrilateral, and let $M$ and$ N$ be midpoints of sides $AB$ and $CD$ respectively. Point $P$ is chosen on $CD$ so that $MP \perp CD$, and point $Q$ is chosen on $AB$ so that $NQ \perp AB$. Show that $AD \parallel BC$ if and only if $\frac{AB}{CD} =\frac{MP}{NQ}$ .
In the triangle $ABC$, points $D, E, F$ are on the sides $BC, CA$ and $AB$ respectively such that $FE$ is parallel to $BC$ and $DF$ is parallel to $CA$, Let P be the intersection of $BE$ and $DF$, and $Q$ the intersection of $FE$ and $AD$. Prove that $PQ$ is parallel to $AB$.
In a triangle $ABC$, $\angle ABC= 60^o$, point $I$ is the incenter. Let the points $P$ and $T$ on the sides $AB$ and $BC$ respectively such that $PI \parallel BC$ and $TI \parallel AB$ , and points $P_1$ and $T_1$ on the sides $AB$ and $BC$ respectively such that $AP_1 = BP$ and $CT_1 = BT$. Prove that point $I$ lies on segment $P_1T_1$.
(Anton Trygub)
Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.
In two circles intersecting at points $A$ and $B$, parallel chords $A_1B_1$ and $A_2B_2$ are drawn. The lines $AA_1$ and $BB_2$ intersect at the point $X, AA_2$ and $BB_1$ intersect at the point $Y$. Prove that $XY // A_1B_1$.
Let $ABC$ be an equilateral triangle. The points $M$ and $K$ are located in different half-planes with respect to line $BC$, so that the point $M \in (AB)$ ¸and the triangle $MKC$ is equilateral. Prove that the lines $AC$ and $BK$ are parallel.
Two equal circles are internally tangent to a larger circle at $A$ and $B$. Let $M$ be a point on the larger circle, and let lines $MA$ and $MB$ intersect the corresponding smaller circles at $A'$ and $B'$. Prove that $A'B'$ is parallel to $AB$.
All vertices of triangles $ABC$ and $A_1B_1C_1$ lie on the hyperbola $y=1/x$. It is known that $AB \parallel A_1B_1$ and $BC \parallel B_1C_1$. Prove that $AC_1 \parallel A_1C$.
(I. Gorodnin)
In the coordinate space, each of the eight vertices of a rectangular box has integer coordinates. If the volume of the solid is $2011$, prove that the sides of the rectangular box are parallel to the coordinate axes.