Found problems: 844
2010 ELMO Shortlist, 2
Given a triangle $ABC$, a point $P$ is chosen on side $BC$. Points $M$ and $N$ lie on sides $AB$ and $AC$, respectively, such that $MP \parallel AC$ and $NP \parallel AB$. Point $P$ is reflected across $MN$ to point $Q$. Show that triangle $QMB$ is similar to triangle $CNQ$.
[i]Brian Hamrick.[/i]
Estonia Open Junior - geometry, 1996.1.4
In a trapezoid, the two non parallel sides and a base have length $1$, while the other base and both the diagonals have length $a$. Find the value of $a$.
2016 JBMO Shortlist, 3
A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.
2011 Regional Competition For Advanced Students, 3
Let $k$ be a circle centered at $M$ and let $t$ be a tangentline to $k$ through some point $T\in k$. Let $P$ be a point on $t$ and let $g\neq t$ be a line through $P$ intersecting $k$ at $U$ and $V$. Let $S$ be the point on $k$ bisecting the arc $UV$ not containing $T$ and let $Q$ be the the image of $P$ under a reflection over $ST$.
Prove that $Q$, $T$, $U$ and $V$ are vertices of a trapezoid.
1990 Tournament Of Towns, (260) 4
Let $ABCD$ be a trapezium with $AC = BC$. Let $H$ be the midpoint of the base $AB$ and let $\ell$ be a line passing through $H$. Let $\ell$ meet $AD$ at $P$ and $BD$ at $Q$. Prove that the angles $ACP$ and $QCB$ are either equal or have a sum of $180^o$.
(I. Sharygin, Moscow)
2006 Harvard-MIT Mathematics Tournament, 7
Suppose $ABCD$ is an isosceles trapezoid in which $\overline{AB}\parallel\overline{CD}$. Two mutually externally tangent circles $\omega_1$ and $\omega_2$ are inscribed in $ABCD$ such that $\omega_1$ is tangent to $\overline{AB}$,$\overline{BC}$, and $\overline{CD}$ while $\omega_2$ is tangent to $\overline{AB}$, $\overline{DA}$, and $\overline{CD}$. Given that $AB=1$, $CD=6$, compute the radius of either circle.
2012 Tuymaada Olympiad, 3
Point $P$ is taken in the interior of the triangle $ABC$, so that
\[\angle PAB = \angle PCB = \dfrac {1} {4} (\angle A + \angle C).\]
Let $L$ be the foot of the angle bisector of $\angle B$. The line $PL$ meets the circumcircle of $\triangle APC$ at point $Q$. Prove that $QB$ is the angle bisector of $\angle AQC$.
[i]Proposed by S. Berlov[/i]
2009 Belarus Team Selection Test, 3
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
1989 AMC 8, 15
The area of the shaded region $\text{BEDC}$ in parallelogram $\text{ABCD}$ is
[asy]
unitsize(10);
pair A,B,C,D,E;
A=origin; B=(4,8); C=(14,8); D=(10,0); E=(4,0);
draw(A--B--C--D--cycle);
fill(B--E--D--C--cycle,gray);
label("A",A,SW); label("B",B,NW); label("C",C,NE); label("D",D,SE); label("E",E,S);
label("$10$",(9,8),N); label("$6$",(7,0),S); label("$8$",(4,4),W);
draw((3,0)--(3,1)--(4,1));
[/asy]
$\text{(A)}\ 24 \qquad \text{(B)}\ 48 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 80$
2008 Germany Team Selection Test, 2
The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$.
[i]Author: Vyacheslav Yasinskiy, Ukraine[/i]
2002 India IMO Training Camp, 13
Let $ABC$ and $PQR$ be two triangles such that
[list]
[b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.
[b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$
[/list]
Prove that $AB+AC=PQ+PR$.
Novosibirsk Oral Geo Oly IX, 2017.6
In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.
2005 Rioplatense Mathematical Olympiad, Level 3, 2
In trapezoid $ABCD$, the sum of the lengths of the bases $AB$ and $CD$ is equal to the length of the diagonal $BD$. Let $M$ denote the midpoint of $BC$, and let $E$ denote the reflection of $C$ about the line $DM$. Prove that $\angle AEB=\angle ACD$.
2013 AMC 12/AHSME, 13
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA=\angle DCB$ and $\angle ADB=\angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$?
${\textbf{(A)}\ 210\qquad\textbf{(B)}\ 220\qquad\textbf{(C)}\ 230\qquad\textbf{(D}}\ 240\qquad\textbf{(E)}\ 250$
2010 Korea - Final Round, 4
Given is a trapezoid $ ABCD$ where $ AB$ and $ CD$ are parallel, and $ A,B,C,D$ are clockwise in this order. Let $ \Gamma_1$ be the circle with center $ A$ passing through $ B$, $ \Gamma_2$ be the circle with center $ C$ passing through $ D$. The intersection of line $ BD$ and $ \Gamma_1$ is $ P$ $ ( \ne B,D)$. Denote by $ \Gamma$ the circle with diameter $ PD$, and let $ \Gamma$ and $ \Gamma_1$ meet at $ X$$ ( \ne P)$. $ \Gamma$ and $ \Gamma_2$ meet at $ Y$. If the circumcircle of triangle $ XBY$ and $ \Gamma_2$ meet at $ Q$, prove that $ B,D,Q$ are collinear.
2005 Purple Comet Problems, 18
The side lengths of a trapezoid are $\sqrt[4]{3}, \sqrt[4]{3}, \sqrt[4]{3}$, and $2 \cdot \sqrt[4]{3}$. Its area is the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$.
2009 Stanford Mathematics Tournament, 7
An isosceles trapezoid has legs and shorter base of length $1$. Find the maximum possible value of its area
2024 Sharygin Geometry Olympiad, 9
Let $ABCD$ ($AD \parallel BC$) be a trapezoid circumscribed around a circle $\omega$, which touches the sides $AB, BC, CD, $ and $AD$ at points $P, Q, R, S$ respectively. The line passing through $P$ and parallel to the bases of the trapezoid meets $QR$ at point $X$. Prove that $AB, QS$ and $DX$ concur.
2017 All-Russian Olympiad, 2
$ABCD$ is an isosceles trapezoid with $BC || AD$. A circle $\omega$ passing through $B$ and $C$ intersects the side $AB$ and the diagonal $BD$ at points $X$ and $Y$ respectively. Tangent to $\omega$ at $C$ intersects the line $AD$ at $Z$. Prove that the points $X$, $Y$, and $Z$ are collinear.
2013 International Zhautykov Olympiad, 1
Given a trapezoid $ABCD$ ($AD \parallel BC$) with $\angle ABC > 90^\circ$ . Point $M$ is chosen on the lateral side $AB$. Let $O_1$ and $O_2$ be the circumcenters of the triangles $MAD$ and $MBC$, respectively. The circumcircles of the triangles $MO_1D$ and $MO_2C$ meet again at the point $N$. Prove that the line $O_1O_2$ passes through the point $N$.
2024 Harvard-MIT Mathematics Tournament, 5
Let $ABCD$ be a convex trapezoid such that $\angle{DAB}=\angle{ABC}=90^{\circ},DA=2,AB=3,$ and $BC=8$. Let $\omega$ be a circle passing through $A$ and tangent to segment $CD$ at point $T$. Suppose that the center of $\omega$ lies on line $BC$. Compute $CT$.
2005 IMO Shortlist, 6
Let $ABC$ be a triangle, and $M$ the midpoint of its side $BC$. Let $\gamma$ be the incircle of triangle $ABC$. The median $AM$ of triangle $ABC$ intersects the incircle $\gamma$ at two points $K$ and $L$. Let the lines passing through $K$ and $L$, parallel to $BC$, intersect the incircle $\gamma$ again in two points $X$ and $Y$. Let the lines $AX$ and $AY$ intersect $BC$ again at the points $P$ and $Q$. Prove that $BP = CQ$.
2009 Iran MO (3rd Round), 3
3-There is given a trapezoid $ ABCD$ in the plane with $ BC\parallel{}AD$.We know that the angle bisectors of the angles of the trapezoid are concurrent at $ O$.Let $ T$ be the intersection of the diagonals $ AC,BD$.Let $ Q$ be on $ CD$ such that $ \angle OQD \equal{} 90^\circ$.Prove that if the circumcircle of the triangle $ OTQ$ intersects $ CD$ again at $ P$ then $ TP\parallel{}AD$.
2023 Iranian Geometry Olympiad, 1
All of the polygons in the figure below are regular. Prove that $ABCD$ is an isosceles trapezoid.
[img]https://cdn.artofproblemsolving.com/attachments/e/a/3f4de32becf4a90bf0f0b002fb4d8e724e8844.png[/img]
[i]Proposed by Mahdi Etesamifard - Iran[/i]
2017 Novosibirsk Oral Olympiad in Geometry, 6
In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.