Found problems: 250
In convex hexagon $ABCDEF, AB$ is parallel to $CF, CD$ is parallel to $BE$ and $EF$ is parallel to $AD$. Prove that the areas of triangles $ACE$ and $BDF$ are equal .
Let $ABC$ be a non-right-angled triangle, with $AC\ne BC$. Let $F$ be the midpoint of side $BC$. Let $D$ be a point on line $AB$ satisfying$CA=CD$,and let $E$ be a point on line $BC$ satisfying $EB = ED$. The line passing through $A$ and parallel to $ED$ meets line $FD$ at point $I$. Line $AF$ meets line $ED$ at point $J$. Prove that points $C$, $I$ and $J$ are collinear.
A convex hexagon $ABCDEF$ is given whose sides $AB$ and $DE$ are parallel. Each of the diagonals $AD, BE, CF$ divides this hexagon into two quadrilaterals of equal perimeters. Show that these three diagonals intersect at one point.
In a quadrilateral $ABCD$ points $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$, respectively. The line through $M$ and $N$ meets $AB$ and $CD$ at $M'$ and $N'$, respectively. Prove that if $MM' = NN'$, then $AD // BC$.
In a scalene triangle $ABC$, let the angle bisector of $A$ meets side $BC$ at $D$. Let $E, F$ be the circumcenter of the triangles $ABD$ and $ADC$, respectively. Suppose that the circumcircles of the triangles $BDE$ and $DCF$ intersect at $P(\neq D)$, and denote by $O, X, Y$ the circumcenters of the triangles $ABC, BDE, DCF$, respectively. Prove that $OP$ and $XY$ are parallel.
Given a triangle $ABC$ with side $AB$ greater than $BC$, let $M$ be the midpoint of $AC$ and $L$ be the point at which the bisector of angle $\angle B$ intersects side $AC$. The line parallel to $AB$, which intersects the bisector $BL$ at $D$, is drawn by $M$, and the line parallel to the side $BC$ that intersects the median $BM$ at $E$ is drawn by $L$. Show that $ED$ is perpendicular to $BL$.
The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.
Inside the triangle $ABC$ choose the point $M$, and on the side $BC$ - the point $K$ in such a way that $MK || AB$. The circle passing through the points $M, \, \, K, \, \, C,$ crosses the side $AC$ for the second time at the point $N$, a circle passing through the points $M, \, \, N, \, \, A, $ crosses the side $AB$ for the second time at the point $Q$. Prove that $BM = KQ$.
(Nagel Igor)
A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing exactly two nodes inside. Prove that the straight line connecting these nodes either passes through a vertex or is parallel to a side of the triangle.
In the rectangular parallelepiped $ABCDA'B'C'D'$ we denote by $M$ the center of the face $ABB'A'$. We denote by $M_1$ and $M_2$ the projections of $M$ on the lines $B'C$ and $AD'$ respectively. Prove that:
a) $MM_1 = MM_2$
b) if $(MM_1M_2) \cap (ABC) = d$, then $d \parallel AD$;
c) $\angle (MM_1M_2), (A B C)= 45^ o \Leftrightarrow \frac{BC}{AB}=\frac{BB'}{BC}+\frac{BC}{BB'}$.
The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square.
[img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]
The diagonals of trapezoid $ABCD$ meet at point $O$. Point $M$ of lateral side $CD$ and points $P, Q$ of bases $BC$ and $AD$ are such that segments $MP$ and $MQ$ are parallel to the diagonals of the trapezoid. Prove that line $PQ$ passes through point $O$.
Let $ABC$ be an acute triangle with $AC > AB$ and circumcircle $\Gamma$. The tangent from $A$
to $\Gamma$ intersects $BC$ at $T$. Let $M$ be the midpoint of $BC$ and let $R$ be the reflection of $A$ in $B$.
Let $S$ be a point so that $SABT$ is a parallelogram and finally let $P$ be a point on line $SB$ such
that $MP$ is parallel to $AB$.
Given that $P$ lies on $\Gamma$, prove that the circumcircle of $\triangle STR$ is tangent to line $AC$.
[i]Proposed by Sam Bealing, United Kingdom[/i]
Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD,TC=TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.
Let $ABCD$ be a convex quadrilateral inscibed in circle $(O)$ such that $DB = DA + DC$. The point $P$ lies on the ray $AC$ such that $AP = BC$. The point $E$ is on $(O)$ such that $BE \perp AD$. Prove that $DP$ is parallel to the angle bisector of $\angle BEC$.
Each of Mahdi and Morteza has drawn an inscribed $93$-gon. Denote the first one by $A_1A_2…A_{93}$ and the second by $B_1B_2…B_{93}$. It is known that $A_iA_{i+1} // B_iB_{i+1}$ for $1 \le i \le 93$ ($A_{93} = A_1, B_{93} = B_1$). Show that $\frac{A_iA_{i+1} }{ B_iB_{i+1}}$ is a constant number independent of $i$.
by Morteza Saghafian
Given a triangle $ABC$ with $\angle C = 90^o$.
a) Construct the circle with center $C$, so that one of the tangents from $A$ to that circle is parallel to one of the tangents from $B$ to that circle.
b) A circle with center $C$ has two parallel tangents passing through A and go respectively. If $AC = b$ and $BC = a$, express the radius of the circle in terms of $a$ and $b$.
Let $ABC$ be an acute triangle with $AB < AC$ and inscribed in the circle $(O)$. Denote $I$ as the incenter of $ABC$ and $D$, $E$ as the intersections of $AI$ with $BC$, $(O)$ respectively. Take a point $K$ on $BC$ such that $\angle AIK = 90^o$ and $KA$, $KE$ meet $(O)$ again at M,N respectively. The rays $ND$, $NI$ meet the circle $(O)$ at $Q$,$P$.
1. Prove that the quadrilateral $MPQE$ is a kite.
2. Take $J$ on $IO$ such that $AK \perp AJ$. The line through $I$ and perpendicular to $OI$ cuts $BC$ at $R$ ,cuts $EK$ at $S$ .Prove that $OR \parallel JS$.
The bisectors of angles $A$ and $C$ of triangle $ABC$ intersect its sides at points $A_1$ and $C_1$, and the circumcircle of this triangle is at points $A_0$ and $C_0$, respectively. Lines $A_1C_1$ and $A_0C_0$ intersect at point P. Prove that the segment connecting $P$ to the center of the incircle of triangle $ABC$ is parallel to $AC$.
Given trapezoid $ABCD$ ($AD\parallel BC$) with $AD \perp AB$ and $T=AC\cap BD$. A circle centered at point $O$ is inscribed in the trapezoid and touches the side $CD$ at point $Q$. Let $P$ be the intersection point (different from $Q$) of the side $CD$ and the circle passing through $T,Q$ and $O$. Prove that $TP \parallel AD$.
I. Voronovich
In a trapezoid $ABCD$ with bases $AD$ and $BC$, the bisector of the angle $\angle DAB$ intersects the bisectors of the angles $\angle ABC$ and $\angle CDA$ at the points $P$ and $S$, respectively, and the bisector of the angle $\angle BCD$ intersects the bisectors of the angles $\angle ABC$ and $\angle CDA$ at the points $Q$ and $R$, respectively. Prove that if $PS\parallel RQ$, then $AB = CD$.
Let $A, B, C, D, E$ be five distinct points on a circle $\Gamma$ in the clockwise order and let the extensions of $CD$ and $AE$ meet at a point $Y$ outside $\Gamma$. Suppose $X$ is a point on the extension of $AC$ such that $XB$ is tangent to $\Gamma$ at $B$. Prove that $XY = XB$ if and only if $XY$ is parallel $DE$.
The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square.
[img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]
A circle concentric with the inscribed circle of $ABC$ intersects the sides of the triangle at six points forming a convex hexagon $A_1A_2B_1B_2C_1C_2$ (points $C_1$ and $C_2$ on the $AB$ side, $A_1$ and $A_2$ on $BC$, $B_1$ and $B_2$ on $AC$). Prove that if line $A_1B_1$ is parallel to the bisector of angle $B$, then line $A_2C_2$ is parallel to the bisector of angle $C$.
A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing at least two nodes inside. Prove that there exists a pair of internal nodes such that a straight line connecting them either passes through a vertex or is parallel to side of the triangle.