Found problems: 844
Let $ABCD$ be a trapezoid with bases $BC \parallel AD$, where $AD > BC$, and non-parallel legs $AB$ and $CD$. Let $M$ be the intersection of $AC$ and $BD$. Let $\Gamma_1$ be a circumference that passes through $M$ and is tangent to $AD$ at point $A$; let $\Gamma_2$ be a circumference that passes through $M$ and is tangent to $AD$ at point $D$. Let $S$ be the intersection of the lines $AB$ and $CD$, $X$ the intersection of $\Gamma_1$ with the line $AS$, $Y$ the intesection of $\Gamma_2$ with the line $DS$, and $O$ the circumcenter of triangle $ASD$.
Show that $SO \perp XY$.
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 $F$.
Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.
A truncated cone has horizontal bases with radii $ 18$ and $ 2$. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 4\sqrt5 \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 6\sqrt3$
An isosceles trapezium is called [i]right[/i] if only one pair of its sides are parallel (i.e parallelograms are not right).
A dissection of a rectangle into $n$ (can be different shapes) right isosceles trapeziums is called [i]strict[/i] if the union of any $i,(2\leq i \leq n)$ trapeziums in the dissection do not form a right isosceles trapezium.
Prove that for any $n, n\geq 9$ there is a strict dissection of a $2017 \times 2018$ rectangle into $n$ right isosceles trapeziums.
[i]Proposed by Bojan Basic[/i]
Let $ABCD$ be an isosceles trapezoid with base $BC$ and $AD$. Suppose $\angle BDC=10^{\circ}$ and $\angle BDA=70^{\circ}$. Show that $AD^2=BC(AD+AB)$.
For what positive integers $n$ is it possible to tile an equilateral triangle of side $n$ with trapezoids each of which has sides $1, 1, 1, 2$?
(NB Vassiliev)
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)
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$.
Prove the following inequality.
\[ \frac{2}{2\plus{}e^{\frac 12}}<\int_0^1 \frac{dx}{1\plus{}xe^{x}}<\frac{2\plus{}e}{2(1\plus{}e)}\]
In a triangle $ABC$, let $D$ be the point on the segment $BC$ such that $AB+BD=AC+CD$. Suppose that the points $B$, $C$ and the centroids of triangles $ABD$ and $ACD$ lie on a circle. Prove that $AB=AC$.
Given is a tetrahedron $ABCD$ in which $AB=CD$ and the sum of measures of the angles $BAD$ and $BCD$ equals $180$ degrees. Prove that the measure of the angle $BAD$ is larger than the measure of the angle $ADC$.
Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
In trapezium $ABCD$, $BC \perp AB$, $BC\perp CD$, and $AC\perp BD$. Given $AB=\sqrt{11}$ and $AD=\sqrt{1001}$. Find $BC$
In a trapezoid $ABCD$ with base $AB$ let $E$ be the midpoint of side $AD$. Suppose further that $2CD=EC=BC=b$. Let $\angle ECB=120^{\circ}$. Construct the trapezoid and determine its area based on $b$.
What is the largest possible area of an isosceles trapezoid in which the largest side is $13$ and the perimeter is $28$?
$
\textbf{(A)}\ 13
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 27
\qquad\textbf{(D)}\ 28
\qquad\textbf{(E)}\ 30
$
Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid?
[asy]
pair A,B,C,D;
A=(3,20);
B=(35,20);
C=(47,0);
D=(0,0);
draw(A--B--C--D--cycle);
dot((0,0));
dot((3,20));
dot((35,20));
dot((47,0));
label("A",A,N);
label("B",B,N);
label("C",C,S);
label("D",D,S);
draw((19,20)--(19,0));
dot((19,20));
dot((19,0));
draw((19,3)--(22,3)--(22,0));
label("12",(21,10),E);
label("50",(19,22),N);
label("15",(1,10),W);
label("20",(41,12),E);[/asy]
$ \textbf{(A)}600\qquad\textbf{(B)}650\qquad\textbf{(C)}700\qquad\textbf{(D)}750\qquad\textbf{(E)}800 $
Two circles are tangent externaly at a point $B$. A line tangent to one of the circles at a point $A$ intersects the other circle at points $C$ and $D$. Show that $A$ is equidistant to the lines $BC$ and $BD$.
A terrain ( $ABCD$ ) has a rectangular trapezoidal shape. The angle in $A$ measures $90^o$. $AB$ measures $30$ m, $AD$ measures $20$ m and $DC$ measures 45 m. This land must be divided into two areas of the same area, drawing a parallel to the $AD$ side . At what distance from $D$ do we have to draw the parallel?
[img]https://1.bp.blogspot.com/-DnyNY3x4XKE/XNYvRUrLVTI/AAAAAAAAKLE/gohd7_S9OeIi-CVUVw-iM63uXE5u-WmGwCK4BGAYYCw/s400/image002.gif[/img]
A square $ABCD$ is given. A point $P$ is chosen inside the triangle $ABC$ such that $\angle CAP = 15^\circ = \angle BCP$. A point $Q$ is chosen such that $APCQ$ is an isosceles trapezoid: $PC \parallel AQ$, and $AP=CQ, AP\nparallel CQ$. Denote by $N$ the midpoint of $PQ$. Find the angles of the triangle $CAN$.
Let $ ABCD$ be a trapezoid such that $ AB \parallel CD$ and assume that there are points $ E$ on the line outside the segment $ BC$ and $ F$ on the segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Let $ I,J,K$ respectively be the intersection of line $ EF$ and line $ CD$, the intersection of line $ EF$ and line $ AB$, and the midpoint of segment $ EF$. Prove that $ K$ is on the circumcircle of triangle $ CDJ$ if and only if $ I$ is on the circumcircle of triangle $ ABK$.
[i]Utari Wijayanti, Bandung[/i]
$ABCD$ is a trapezoid with $AB \parallel CD$. The diagonals intersect at $P$. Let $\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$. Let $\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$. $\omega _3$ is the circumcircle of triangle $BPC$.
Prove that the common chord of circles $\omega _1,\omega _3$ and the common chord of circles $\omega _2, \omega _3$ intersect each other on $AD$.
[i]Proposed by Kasra Ahmadi[/i]
Find the smallest integer $n$ satisfying the following condition: regardless of how one colour the vertices of a regular $n$-gon with either red, yellow or blue, one can always find an isosceles trapezoid whose vertices are of the same colour.
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 $BK = MN$.
Minor base $BC$ of trapezoid $ABCD$ is equal to side $AB$, and diagonal $AC$ is equal to base $AD$. The line passing through B and parallel to $AC$ intersects line $DC$ in point $M$. Prove that $AM$ is the bisector of angle $\angle BAC$.
A.Blinkov, Y.Blinkov