Found problems: 844
2009 Hong Kong TST, 4
Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$.
(a) Show that all such lines $ AB$ are concurrent.
(b) Find the locus of midpoints of all such segments $ AB$.
1972 AMC 12/AHSME, 18
Let $ABCD$ be a trapezoid with the measure of base $AB$ twice that of base $DC$, and let $E$ be the point of intersection of the diagonals. If the measure of diagonal $AC$ is $11$, then that of segment $EC$ is equal to
$\textbf{(A) }3\textstyle\frac{2}{3}\qquad\textbf{(B) }3\frac{3}{4}\qquad\textbf{(C) }4\qquad\textbf{(D) }3\frac{1}{2}\qquad \textbf{(E) }3$
1989 Greece National Olympiad, 4
A trapezoid with bases $a,b$ and altitude $h$ is circumscribed around a circl.. Prove that $h^2\le ab$.
2012 NZMOC Camp Selection Problems, 2
Let $ABCD$ be a trapezoid, with $AB \parallel CD$ (the vertices are listed in cyclic order). The diagonals of this trapezoid are perpendicular to one another and intersect at $O$. The base angles $\angle DAB$ and $\angle CBA$ are both acute. A point $M$ on the line sgement $OA$ is such that $\angle BMD = 90^o$, and a point $N$ on the line segment $OB$ is such that $\angle ANC = 90^o$. Prove that triangles $OMN$ and $OBA$ are similar.
1996 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 3
Let $ ABCD$ be a trapezoid with $ AB$ and $ CD$ parallel, $ \angle D \equal{} 2 \angle B, AD \equal{} 5,$ and $ CD \equal{} 2.$ Then $ AB$ equals
A. 7
B. 8
C. 13/2
D. 27/4
E. $ 5 \plus{} \frac{3 \sqrt{2}}{2}$
2024 All-Russian Olympiad Regional Round, 9.2
On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the diagonals of all such trapezoids share a common point.
2014 Postal Coaching, 4
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 $AB+AC=PQ+PR$.
2011 China Team Selection Test, 1
Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.
2017-2018 SDML (Middle School), 6
In the figure, a circle is located inside a trapezoid with two right angles so that a point of tangency of the circle is the midpoint of the side perpendicular to the two bases. The circle also has points of tangency on each base of the trapezoid. The diameter of the circle is $\frac{2}{3}$ the length of $EF$. If the area of the circle is $9\pi$ square units, what is the area of the trapezoid?
[asy]
draw((0,0) -- (11, 0) -- (7,6) -- (0,6) -- cycle);
draw((0,3) -- (9,3));
draw(circle((3,3), 3));
draw(rightanglemark((1,0),(0,0),(0,1),12));
draw(rightanglemark((0,0),(0,6),(6,6), 12));
label("E", (0,3), W);
label("F", (9,3), E);
[/asy]
Durer Math Competition CD Finals - geometry, 2022.D4
The longer base of trapezoid $ABCD$ is $AB$, while the shorter base is $CD$. Diagonal $AC$ bisects the interior angle at $A$. The interior bisector at $B$ meets diagonal $AC$ at $E$. Line $DE$ meets segment $AB$ at $F$. Suppose that $AD = FB$ and $BC = AF$. Find the interior angles of quadrilateral $ABCD$, if we know that $\angle BEC = 54^o$.
2018 Cyprus IMO TST, 2
Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$.
2015 Costa Rica - Final Round, 6
Given the trapezoid $ABCD$ with the $BC\parallel AD$, let $C_1$ and $C_2$ be circles with diameters $AB$ and $CD$ respectively. Let $M$ and $N$ be the intersection points of $C_1$ with $AC$ and $BD$ respectively. Let $K$ and $L$ be the intersection points of $C_2$ with $AC$ and $BD$ respectively. Given $M\ne A$, $N\ne B$, $K\ne C$, $L\ne D$. Prove that $NK \parallel ML$.
2024 China Team Selection Test, 19
$n$ is a positive integer. An equilateral triangle of side length $3n$ is split into $9n^2$ unit equilateral triangles, each colored one of red, yellow, blue, such that each color appears $3n^2$ times. We call a trapezoid formed by three unit equilateral triangles as a "standard trapezoid". If a "standard trapezoid" contains all three colors, we call it a "colorful trapezoid". Find the maximum possible number of "colorful trapezoids".
2020-21 IOQM India, 1
Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AB = 3CD$. Let $E$ be then midpoint of the diagonal $BD$. If $[ABCD] = n \times [CDE]$, what is the value of $n$?
(Here $[t]$ denotes the area of the geometrical figure$ t$.)
2005 Romania National Olympiad, 3
Let $ABCD$ be a quadrilateral with $AB\parallel CD$ and $AC \perp BD$. Let $O$ be the intersection of $AC$ and $BD$. On the rays $(OA$ and $(OB$ we consider the points $M$ and $N$ respectively such that $\angle ANC = \angle BMD = 90^\circ$. We denote with $E$ the midpoint of the segment $MN$. Prove that
a) $\triangle OMN \sim \triangle OBA$;
b) $OE \perp AB$.
[i]Claudiu-Stefan Popa[/i]
2022 European Mathematical Cup, 4
Five points $A$, $B$, $C$, $D$ and $E$ lie on a circle $\tau$ clockwise in that order such that $AB \parallel CE$ and $\angle ABC > 90^{\circ}$. Let $k$ be a circle tangent to $AD$, $CE$ and $\tau$ such that $k$ and $\tau$ touch on the arc $\widehat{DE}$ not containing $A$, $B$ and $C$. Let $F \neq A$ be the intersection of $\tau$ and the tangent line to $k$ passing through $A$ different from $AD$.
Prove that there exists a circle tangent to $BD$, $BF$, $CE$ and $\tau$.
2018 Oral Moscow Geometry Olympiad, 2
Bisectors of angle $C$ and externalangle $A$ of trapezoid $ABCD$ with bases $BC$ and $AD$ intersect at point $M$, and the bisector of angle $B$ and external angle $D$ intersect at point $N$. Prove that the midpoint of the segment $MN$ is equidistant from the lines $AB$ and $CD$.
2015 Oral Moscow Geometry Olympiad, 4
In trapezoid $ABCD$, the bisectors of angles $A$ and $D$ intersect at point $E$ lying on the side of $BC$. These bisectors divide the trapezoid into three triangles into which the circles are inscribed. One of these circles touches the base $AB$ at the point $K$, and two others touch the bisector $DE$ at points $M$ and $N$. Prove that $BK = MN$.
2022 AMC 12/AHSME, 22
Let $c$ be a real number, and let $z_1, z_2$ be the two complex numbers satisfying the quadratic $z^2 - cz + 10 = 0$. Points $z_1, z_2, \frac{1}{z_1}$, and $\frac{1}{z_2}$ are the vertices of a (convex) quadrilateral $Q$ in the complex plane. When the area of $Q$ obtains its maximum value, $c$ is the closest to which of the following?
$\textbf{(A)}~4.5\qquad\textbf{(B)}~5\qquad\textbf{(C)}~5.5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~6.5$
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 $AB$.
1998 ITAMO, 4
Let $ABCD$ be a trapezoid with the longer base $AB$ such that its diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of the triangle $ABC$ and $E$ be the intersection of the lines $OB$ and $CD$. Prove that $BC^2 = CD \cdot CE$.
2008 AMC 10, 20
Trapezoid $ ABCD$ has bases $ \overline{AB}$ and $ \overline{CD}$ and diagonals intersecting at $ K$. Suppose that $ AB\equal{}9$, $ DC\equal{}12$, and the area of $ \triangle AKD$ is $ 24$. What is the area of trapezoid $ ABCD$?
$ \textbf{(A)}\ 92 \qquad
\textbf{(B)}\ 94 \qquad
\textbf{(C)}\ 96 \qquad
\textbf{(D)}\ 98 \qquad
\textbf{(E)}\ 100$
2021 CCA Math Bonanza, I12
Let $ABC$ be a triangle, let the $A$-altitude meet $BC$ at $D$, let the $B$-altitude meet $AC$ at $E$, and let $T\neq A$ be the point on the circumcircle of $ABC$ such that $AT || BC$. Given that $D,E,T$ are collinear, if $BD=3$ and $AD=4$, then the area of $ABC$ can be written as $a+\sqrt{b}$, where $a$ and $b$ are positive integers. What is $a+b$?
[i]2021 CCA Math Bonanza Individual Round #12[/i]
2007 Ukraine Team Selection Test, 2
$ ABCD$ is convex $ AD\parallel BC$, $ AC\perp BD$. $ M$ is interior point of $ ABCD$ which is not a intersection of diagonals $ AC$ and $ BD$ such that $ \angle AMB \equal{}\angle CMD \equal{}\frac{\pi}{2}$ .$ P$ is intersection of angel bisectors of $ \angle A$ and $ \angle C$. $ Q$ is intersection of angel bisectors of $ \angle B$ and $ \angle D$. Prove that $ \angle PMB \equal{}\angle QMC$.
2003 Tournament Of Towns, 3
Points $K$ and $L$ are chosen on the sides $AB$ and $BC$ of the isosceles $\triangle ABC$ ($AB = BC$) so that $AK +LC = KL$. A line parallel to $BC$ is drawn through midpoint $M$ of the segment $KL$, intersecting side $AC$ at point $N$. Find the value of $\angle KNL$.