Found problems: 844
2007 Iran Team Selection Test, 3
Let $\omega$ be incircle of $ABC$. $P$ and $Q$ are on $AB$ and $AC$, such that $PQ$ is parallel to $BC$ and is tangent to $\omega$. $AB,AC$ touch $\omega$ at $F,E$. Prove that if $M$ is midpoint of $PQ$, and $T$ is intersection point of $EF$ and $BC$, then $TM$ is tangent to $\omega$.
[i]By Ali Khezeli[/i]
2013 Online Math Open Problems, 36
Let $ABCD$ be a nondegenerate isosceles trapezoid with integer side lengths such that $BC \parallel AD$ and $AB=BC=CD$. Given that the distance between the incenters of triangles $ABD$ and $ACD$ is $8!$, determine the number of possible lengths of segment $AD$.
[i]Ray Li[/i]
2009 South East Mathematical Olympiad, 2
In the convex pentagon $ABCDE$ we know that $AB=DE, BC=EA$ but $AB \neq EA$.
$B,C,D,E$ are concyclic .
Prove that $A,B,C,D$ are concyclic if and only if $AC=AD.$
2022 Sharygin Geometry Olympiad, 8.8
An isosceles trapezoid $ABCD$ ($AB = CD$) is given. A point $P$ on its circumcircle is such that segments $CP$ and $AD$ meet at point $Q$. Let $L$ be tha midpoint of$ QD$. Prove that the diagonal of the trapezoid is not greater than the sum of distances from the midpoints of the lateral sides to ana arbitrary point of line $PL$.
Swiss NMO - geometry, 2020.7
Let $ABCD$ be an isosceles trapezoid with bases $AD> BC$. Let $X$ be the intersection of the bisectors of $\angle BAC$ and $BC$. Let $E$ be the intersection of$ DB$ with the parallel to the bisector of $\angle CBD$ through $X$ and let $F$ be the intersection of $DC$ with the parallel to the bisector of $\angle DCB$ through $X$. Show that quadrilateral $AEFD$ is cyclic.
2009 Iran MO (3rd Round), 2
2-There is given a trapezoid $ ABCD$.We have the following properties:$ AD\parallel{}BC,DA \equal{} DB \equal{} DC,\angle BCD \equal{} 72^\circ$. A point $ K$ is taken on $ BD$ such that $ AD \equal{} AK,K \neq D$.Let $ M$ be the midpoint of $ CD$.$ AM$ intersects $ BD$ at $ N$.PROVE $ BK \equal{} ND$.
2022 Auckland Mathematical Olympiad, 3
Point $E$ is the midpoint of the base $AD$ of the trapezoid $ABCD$. Segments $BD$ and $CE$ intersect at point $F$. It is known that $AF$ is perpendicular to $BD$. Prove that $BC = FC$.
Ukraine Correspondence MO - geometry, 2005.7
Let $O$ be the point of intersection of the diagonals of the trapezoid $ABCD$ with the bases $AB$ and $CD$. It is known that $\angle AOB = \angle DAB = 90^o$. On the sides $AD$ and $BC$ take the points $E$ and $F$ so that $EF\parallel AB$ and $EF = AD$. Find the angle $\angle AOE$.
1959 IMO Shortlist, 6
Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is given in the plane $P$, and the point $C$ in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB \parallel CD$) in which a circle can be inscribed, and with vertices $B$ and $D$ lying in planes $P$ and $Q$ respectively.
1998 IMO Shortlist, 2
Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.
2024 Korea Junior Math Olympiad (First Round), 16.
There is an Equilateral trapezoid $ ABCD. $
$ \bar{AB} =60, \bar{BC}=\bar{DA}= 36, \bar{CD}=108. $
$ M $ is the middle point of $ \bar {AB} $, and point $P$ on $ \bar{AM} $ follows that $ \bar {AP} $ =10.
The foot of perpendicular dropped from $P$ to $ \bar {BD} $ is $E$.
$ \bar{AC} \cap \bar{BD} $ is $ F $.
Point $X$ is on $ \bar {AF} $ which follows $ \bar{MX}=\bar{ME} $
Find $ \bar{AX} \times \bar{AF} $
1992 IberoAmerican, 2
Given a circle $\Gamma$ and the positive numbers $h$ and $m$, construct with straight edge and compass a trapezoid inscribed in $\Gamma$, such that it has altitude $h$ and the sum of its parallel sides is $m$.
2003 USA Team Selection Test, 6
Let $\overline{AH_1}, \overline{BH_2}$, and $\overline{CH_3}$ be the altitudes of an acute scalene triangle $ABC$. The incircle of triangle $ABC$ is tangent to $\overline{BC}, \overline{CA},$ and $\overline{AB}$ at $T_1, T_2,$ and $T_3$, respectively. For $k = 1, 2, 3$, let $P_i$ be the point on line $H_iH_{i+1}$ (where $H_4 = H_1$) such that $H_iT_iP_i$ is an acute isosceles triangle with $H_iT_i = H_iP_i$. Prove that the circumcircles of triangles $T_1P_1T_2$, $T_2P_2T_3$, $T_3P_3T_1$ pass through a common point.
2005 Korea - Final Round, 3
In a trapezoid $ABCD$ with $AD \parallel BC , O_{1}, O_{2}, O_{3}, O_{4}$ denote the circles with diameters $AB, BC, CD, DA$, respectively. Show that there exists a circle with center inside the trapezoid which is tangent to all the four circles $O_{1},..., O_{4}$ if and only if $ABCD$ is a parallelogram.
2007 Bulgarian Autumn Math Competition, Problem 12.2
All edges of the triangular pyramid $ABCD$ are equal in length. Let $M$ be the midpoint of $DB$, $N$ is the point on $\overline{AB}$, such that $2NA=NB$ and $N\not\in AB$ and $P$ is a point on the altitude through point $D$ in $\triangle BCD$. Find $\angle MPD$ if the intersection of the pyramid with the plane $(NMP)$ is a trapezoid.
2005 Postal Coaching, 6
Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, and let $E$ be the midpoint of its side $BC$. Suppose we can inscribe a circle into the quadrilateral $ABED$, and that we can inscribe a circle into the quadrilateral $AECD$. Denote $|AB|=a$, $|BC|=b$, $|CD|=c$, $|DA|=d$. Prove that \[a+c=\frac{b}{3}+d;\] \[\frac{1}{a}+\frac{1}{c}=\frac{3}{b}\]
2004 Oral Moscow Geometry Olympiad, 5
Trapezoid $ABCD$ with bases $AB$ and $CD$ is inscribed in a circle. Prove that the quadrilateral formed by orthogonal projections of any point of this circle onto lines $AC, BC, AD$ and $BD$ is inscribed.
2012 Greece Team Selection Test, 4
Let $n=3k$ be a positive integer (with $k\geq 2$). An equilateral triangle is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). We colour the points of the grid with three colours (red, blue and green) such that each two neighboring points have different colour.
Finally, the colour of a "trapezoid" will be the colour of the midpoint of its big base.
Find the number of all "trapezoids" in the grid (not necessarily disjoint) and determine the number of red, blue and green "trapezoids".
2017 Azerbaijan Junior National Olympiad, P4
A Rhombus and an Isosceles trapezoid that has same area is drawn in the same circle's outside. Compare their acute angles \\
(explain your answer)
2020 Sharygin Geometry Olympiad, 4
Let $ABCD$ be an isosceles trapezoid with bases $AB$ and $CD$. Prove that the centroid of triangle $ABD$ lies on $CF$ where $F$ is the projection of $D$ to $AB$.
Novosibirsk Oral Geo Oly VIII, 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 $AD: BC$.
2010 Polish MO Finals, 3
$ABCD$ is a parallelogram in which angle $DAB$ is acute. Points $A, P, B, D$ lie on one circle in exactly this order. Lines $AP$ and $CD$ intersect in $Q$. Point $O$ is the circumcenter of the triangle $CPQ$. Prove that if $D \neq O$ then the lines $AD$ and $DO$ are perpendicular.
2014 AMC 8, 14
Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?
[asy]
size(250);
defaultpen(linewidth(0.8));
pair A=(0,5),B=origin,C=(6,0),D=(6,5),E=(18,0);
draw(A--B--E--D--cycle^^C--D);
draw(rightanglemark(D,C,E,30));
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,S);
label("$D$",D,N);
label("$E$",E,S);
label("$5$",A/2,W);
label("$6$",(A+D)/2,N);
[/asy]
$\textbf{(A) }12\qquad\textbf{(B) }13\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad \textbf{(E) }16$
2004 Romania National Olympiad, 3
Let $ABCD$ be an orthodiagonal trapezoid such that $\measuredangle A = 90^{\circ}$ and $AB$ is the larger base. The diagonals intersect at $O$, $\left( OE \right.$ is the bisector of $\measuredangle AOD$, $E \in \left( AD \right)$ and $EF \| AB$, $F \in \left( BC \right)$. Let $P,Q$ the intersections of the segment $EF$ with $AC,BD$. Prove that:
(a) $EP=QF$;
(b) $EF=AD$.
[i]Claudiu-Stefan Popa[/i]
2001 Mediterranean Mathematics Olympiad, 1
Let $P$ and $Q$ be points on a circle $k$. A chord $AC$ of $k$ passes through the midpoint $M$ of $PQ$. Consider a trapezoid $ABCD$ inscribed in $k$ with $AB \parallel PQ \parallel CD$. Prove that the intersection point $X$ of $AD$ and $BC$ depends only on $k$ and $P,Q.$