Found problems: 844
2011 Turkey MO (2nd round), 5
Let $M$ and $N$ be two regular polygonic area.Define $K(M,N)$ as the midpoints of segments $[AB]$ such that $A$ belong to $M$ and $B$ belong to $N$. Find all situations of $M$ and $N$ such that $K(M,N)$ is a regualr polygonic area too.
1988 AMC 12/AHSME, 6
A figure is an equiangular parallelogram if and only if it is a
$ \textbf{(A)}\ \text{rectangle}\qquad\textbf{(B)}\ \text{regular polygon}\qquad\textbf{(C)}\ \text{rhombus}\qquad\textbf{(D)}\ \text{square}\qquad\textbf{(E)}\ \text{trapezoid} $
2021 Sharygin Geometry Olympiad, 9.6
The diagonals of trapezoid $ABCD$ ($BC\parallel AD$) meet at point $O$. Points $M$ and $N$ lie on the segments $BC$ and $AD$ respectively. The tangent to the circle $AMC$ at $C$ meets the ray $NB$ at point $P$; the tangent to the circle $BND$ at $D$ meets the ray $MA$ at point $R$. Prove that $\angle BOP =\angle AOR$.
2019 Tournament Of Towns, 5
The orthogonal projection of a tetrahedron onto a plane containing one of its faces is a trapezoid of area $1$, which has only one pair of parallel sides.
a) Is it possible that the orthogonal projection of this tetrahedron onto a plane containing another its face is a square of area $1$?
b) The same question for a square of area $1/2019$.
(Mikhail Evdokimov)
2016 Sharygin Geometry Olympiad, 6
The sidelines $AB$ and $CD$ of a trapezoid meet at point $P$, and the diagonals of this trapezoid meet at point $Q$. Point $M$ on the smallest base $BC$ is such that $AM=MD$. Prove that $\angle PMB=\angle QMB$.
2016 Greece Junior Math Olympiad, 3
Let $ABCD$ be a trapezoid ($AD//BC$) with $\angle A=\angle B= 90^o$ and $AD<BC$. Let $E$ be the intersection point of the non parallel sides $AB$ and $CD$, $Z$ be the symmetric point of $A$ wrt line $BC$ and $M$ be the midpoint of $EZ$. If it is given than line $CM$ is perpendicular on line $DZ$, then prove that line $ZC$ is perpendicular on line $EC$.
2022 Moldova EGMO TST, 11
Let there be a trapezoid $ABCD$ with bases $AD$ and $BC$. Points $M$ and $P$ are on sides $AB$ and $CD$ such that $CM$ and $BP$ intersect in $N$ and the pentagon $AMNPD$ is cyclic. Prove that the triangle $ADN$ is isosceles.
2014 All-Russian Olympiad, 2
Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $ \Omega $ is a circle passing through $A,B,C,D$. Let $ \omega $ be the circle passing through $C,D$ and intersecting with $CA,CB$ at $A_1$, $B_1$ respectively. $A_2$ and $B_2$ are the points symmetric to $A_1$ and $B_1$ respectively, with respect to the midpoints of $CA$ and $CB$. Prove that the points $A,B,A_2,B_2$ are concyclic.
[i]I. Bogdanov[/i]
2021 Macedonian Mathematical Olympiad, Problem 3
Let $ABCD$ be a trapezoid with $AD \parallel BC$ and $\angle BCD < \angle ABC < 90^\circ$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$. The circumcircle $\omega$ of $\triangle BEC$ intersects the segment $CD$ at $X$. The lines $AX$ and $BC$ intersect at $Y$, while the lines $BX$ and $AD$ intersect at $Z$. Prove that the line $EZ$ is tangent to $\omega$ iff the line $BE$ is tangent to the circumcircle of $\triangle BXY$.
2007 International Zhautykov Olympiad, 3
Let $ABCDEF$ be a convex hexagon and it`s diagonals have one common point $M$. It is known that the circumcenters of triangles $MAB,MBC,MCD,MDE,MEF,MFA$ lie on a circle.
Show that the quadrilaterals $ABDE,BCEF,CDFA$ have equal areas.
2011 Oral Moscow Geometry Olympiad, 3
A non-isosceles trapezoid $ABCD$ ($AB // CD$) is given. An arbitrary circle passing through points $A$ and $B$ intersects the sides of the trapezoid at points $P$ and $Q$, and the intersect the diagonals at points $M$ and $N$. Prove that the lines $PQ, MN$ and $CD$ are concurrent.
1968 IMO Shortlist, 8
Given an oriented line $\Delta$ and a fixed point $A$ on it, consider all trapezoids $ABCD$ one of whose bases $AB$ lies on $\Delta$, in the positive direction. Let $E,F$ be the midpoints of $AB$ and $CD$ respectively. Find the loci of vertices $B,C,D$ of trapezoids that satisfy the following:
[i](i) [/i] $|AB| \leq a$ ($a$ fixed);
[i](ii) [/i] $|EF| = l$ ($l$ fixed);
[i](iii)[/i] the sum of squares of the nonparallel sides of the trapezoid is constant.
[hide="Remark"]
[b]Remark.[/b] The constants are chosen so that such trapezoids exist.[/hide]
JBMO Geometry Collection, 2002
Two circles with centers $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ such that the centers of the circles are on opposite sides of the line $AB$. The lines $BO_{1}$ and $BO_{2}$ meet their respective circles again at $B_{1}$ and $B_{2}$. Let $M$ be the midpoint of $B_{1}B_{2}$. Let $M_{1}$, $M_{2}$ be points on the circles of centers $O_{1}$ and $O_{2}$ respectively, such that $\angle AO_{1}M_{1}= \angle AO_{2}M_{2}$, and $B_{1}$ lies on the minor arc $AM_{1}$ while $B$ lies on the minor arc $AM_{2}$. Show that $\angle MM_{1}B = \angle MM_{2}B$.
[i]Ciprus[/i]
2007 Kyiv Mathematical Festival, 2
The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$
2021 Indonesia TST, G
Given points $A$, $B$, $C$, and $D$ on circle $\omega$ such that lines $AB$ and $CD$ intersect on point $T$ where $A$ is between $B$ and $T$, moreover $D$ is between $C$ and $T$. It is known that the line passing through $D$ which is parallel to $AB$ intersects $\omega$ again on point $E$ and line $ET$ intersects $\omega$ again on point $F$. Let $CF$ and $AB$ intersect on point $G$, $X$ be the midpoint of segment $AB$, and $Y$ be the reflection of point $T$ to $G$.
Prove that $X$, $Y$, $C$, and $D$ are concyclic.
2018 India PRMO, 5
Let $ABCD$ be a trapezium in which $AB //CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$.
2002 AMC 10, 17
A regular octagon $ ABCDEFGH$ has sides of length two. Find the area of $ \triangle{ADG}$.
$ \textbf{(A)}\ 4 \plus{} 2 \sqrt{2} \qquad
\textbf{(B)}\ 6 \plus{} \sqrt{2} \qquad
\textbf{(C)}\ 4 \plus{} 3 \sqrt{2} \qquad
\textbf{(D)}\ 3 \plus{} 4 \sqrt{2} \qquad
\textbf{(E)}\ 8 \plus{} \sqrt{2}$
2006 May Olympiad, 4
Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ . Let $O$ be the intersection point of the diagonals $AC$ and $BD$. If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$, calculate the area of the triangle $BCO$.
2019 BMT Spring, 7
Points $ A, B, C, D $ are vertices of an isosceles trapezoid, with $ \overline{AB} $ parallel to $ \overline{CD} $, $ AB = 1 $, $ CD = 2 $, and $ BC = 1 $. Point $ E $ is chosen uniformly and at random on $ \overline{CD} $, and let point $ F $ be the point on $ \overline{CD} $ such that $ EC = FD $. Let $ G $ denote the intersection of $ \overline{AE} $ and $ \overline{BF} $, not necessarily in the trapezoid. What is the probability that $ \angle AGB > 30^\circ $?
2022-2023 OMMC, 16
Let $ABCD$ be an isosceles trapezoid with $AB=5$, $CD = 8$, and $BC = DA = 6$. There exists an angle $\theta$ such that there is only one point $X$ satisfying $\angle AXD = 180^{\circ} - \angle BXC = \theta$. Find $\sin(\theta)^2$.
2007 Vietnam National Olympiad, 3
Let ABCD be trapezium that is inscribed in circle (O) with larger edge BC. P is a point lying outer segment BC. PA cut (O) at N(that means PA isn't tangent of (O)), the circle with diameter PD intersect (O) at E, DE meet BC at N. Prove that MN always pass through a fixed point.
2003 Balkan MO, 2
Let $ABC$ be a triangle, and let the tangent to the circumcircle of the triangle $ABC$ at $A$ meet the line $BC$ at $D$. The perpendicular to $BC$ at $B$ meets the perpendicular bisector of $AB$ at $E$. The perpendicular to $BC$ at $C$ meets the perpendicular bisector of $AC$ at $F$. Prove that the points $D$, $E$ and $F$ are collinear.
[i]Valentin Vornicu[/i]
2018 AIME Problems, 7
Triangle $ABC$ has sides $AB=9,BC = 5\sqrt{3},$ and $AC=12$. Points $A=P_0, P_1, P_2, \dots, P_{2450} = B$ are on segment $\overline{AB}$ with $P_k$ between $P_{k-1}$ and $P_{k+1}$ for $k=1,2,\dots,2449$, and points $A=Q_0, Q_1, Q_2, \dots ,Q_{2450} = C$ for $k=1,2,\dots,2449$. Furthermore, each segment $\overline{P_kQ_k}, k=1,2,\dots,2449$, is parallel to $\overline{BC}$. The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions have the same area. Find the number of segments $\overline{P_kQ_k}, k=1,2 ,\dots,2450$, that have rational length.
2020 Austrian Junior Regional Competition, 3
Given is an isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB> CD$. The projection from $D$ on $ AB$ is $E$. The midpoint of the diagonal $BD$ is $M$. Prove that $EM$ is parallel to $AC$.
(Karl Czakler)
2014 India Regional Mathematical Olympiad, 1
let $ABCD$ be a isosceles trapezium having an incircle with $AB$ parallel to $CD$.
let $CE$ be the perpendicular from $C$ on $AB$
prove that
$ CE^2 = AB. CD $