Found problems: 844
Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $D$,$E$,$F$ be the points where its incircle meets $BC$, $AC$, $AB$ respectively. Let the circumcircles of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent.
[i]Proposed by Michael Kural[/i]
Let $ ABCDE$ be a convex pentagon such that $ AB\plus{}CD\equal{}BC\plus{}DE$ and $ k$ a circle with center on side $ AE$ that touches the sides $ AB$, $ BC$, $ CD$ and $ DE$ at points $ P$, $ Q$, $ R$ and $ S$ (different from vertices of the pentagon) respectively. Prove that lines $ PS$ and $ AE$ are parallel.
Given quadrilateral $ABCD$. Its sidelines$ AB$ and $CD$ intersect in point $K$. It's diagonals intersect in point $L$. It is known that line $KL$ pass through the centroid of $ABCD$. Prove that $ABCD$ is trapezoid.
(F.Nilov)
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 $?
In a triangle $ABC,$ the internal and external bisectors of the angle $A$ intersect the line $BC$ at $D$ and $E$ respectively. The line $AC$ meets the circle with diameter $DE$ again at $F.$ The tangent line to the circle $ABF$ at $A$ meets the circle with diameter $DE$ again at $G.$ Show that $AF = AG.$
In trapezoid $ ABCD$, $ \overline{AB}$ and $ \overline{CD}$ are perpendicular to $ \overline{AD}$, with $ AB\plus{}CD\equal{}BC$, $ AB<CD$, and $ AD\equal{}7$. What is $ AB\cdot CD$?
$ \textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 12.25 \qquad
\textbf{(C)}\ 12.5 \qquad
\textbf{(D)}\ 12.75 \qquad
\textbf{(E)}\ 13$
Let $ ABCD$ be a trapezoid with $ AB\parallel{}CD$, $ AB\equal{}11$, $ BC\equal{}5$, $ CD\equal{}19$, and $ DA\equal{}7$. Bisectors of $ \angle A$ and $ \angle D$ meet at $ P$, and bisectors of $ \angle B$ and $ \angle C$ meet at $ Q$. What is the area of hexagon $ ABQCDP$?
$ \textbf{(A)}\ 28\sqrt{3}\qquad
\textbf{(B)}\ 30\sqrt{3}\qquad
\textbf{(C)}\ 32\sqrt{3}\qquad
\textbf{(D)}\ 35\sqrt{3}\qquad
\textbf{(E)}\ 36\sqrt{3}$
A trapezoid $ABCD$ is given with $BC // AD$. Assume that the bisectors of the angles $BAD$ and $CDA$ intersect on the perpendicular bisector of the line segment $BC$. Prove that $|AB|= |CD|$ or $|AB| +|CD| =|AD|$.
Consider a trapezium $ ABCD $ in which $ AB\parallel CD. $ Show that
$$ (AC^2+AB^2-BC^2)(BD^2-BC^2+CD^2) =(AC^2-AD^2+CD^2)(BD^2+AB^2-AD^2) . $$
In the trapezoid $ABCD, AB // CD$ and the diagonals intersect at $O$. The points $P, Q$ are on $AD, BC$ respectively such that $\angle AP B = \angle CP D$ and $\angle AQB = \angle CQD$. Show that $OP = OQ$.
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]
Let $A_0A_1A_2A_3A_4A_5$ be a convex hexagon inscribed in a circle. Define the points $A_0'$, $A_2'$, $A_4'$ on the circle, such that
\[ A_0A_0' \parallel A_2A_4, \quad A_2A_2' \parallel A_4A_0, \quad A_4A_4' \parallel A_2A_0 . \]
Let the lines $A_0'A_3$ and $A_2A_4$ intersect in $A_3'$, the lines $A_2'A_5$ and $A_0A_4$ intersect in $A_5'$ and the lines $A_4'A_1$ and $A_0A_2$ intersect in $A_1'$.
Prove that if the lines $A_0A_3$, $A_1A_4$ and $A_2A_5$ are concurrent then the lines $A_0A_3'$, $A_4A_1'$ and $A_2A_5'$ are also concurrent.
We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .
Convex quadrilateral $ ABCD$ has $ AB\equal{}9$ and $ CD\equal{}12$. Diagonals $ AC$ and $ BD$ intersect at $ E$, $ AC\equal{}14$, and $ \triangle AED$ and $ \triangle BEC$ have equal areas. What is $ AE$?
$ \textbf{(A)}\ \frac{9}{2}\qquad \textbf{(B)}\ \frac{50}{11}\qquad \textbf{(C)}\ \frac{21}{4}\qquad \textbf{(D)}\ \frac{17}{3}\qquad \textbf{(E)}\ 6$
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 $O_1 E O_2$.
A triangle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is $ 18$ inches, the median of the trapezoid is:
$ \textbf{(A)}\ 36\text{ inches} \qquad\textbf{(B)}\ 9\text{ inches} \qquad\textbf{(C)}\ 18\text{ inches}\\
\textbf{(D)}\ \text{not obtainable from these data} \qquad\textbf{(E)}\ \text{none of these}$
Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$, $AB=16$, $CD=12$, and $BC<AD$. A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$.
Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$. What is $\tfrac{AB}{BC}$?
$\textbf{(A) \ } 1+\sqrt{2} \qquad \textbf{(B) \ } 2+\sqrt{2}\qquad \textbf{(C) \ } \sqrt{17}\qquad \textbf{(D) \ } 2+\sqrt{5} \qquad \textbf{(E) \ } 1+2\sqrt{3}$
A triangular piece of paper of area $1$ is folded along a line parallel to one of the sides and pressed flat. What is the minimum possible area of the resulting figure?
Points $A,B,C,D,E$ lie on a circle $\omega$ and point $P$ lies outside the circle. The given points are such that (i) lines $PB$ and $PD$ are tangent to $\omega$, (ii) $P, A, C$ are collinear, and (iii) $DE \parallel AC$. Prove that $BE$ bisects $AC$.
In the trapezium $ABCD$ the sides $AB$ and $CD$ are parallel, and $E$ is a fixed point on the side $AB$. Determine the point $F$ on the side $CD$ so that the area of the intersection of the triangles $ABF$ and $CDE$ is as large as possible.
One base of a trapezoid is 100 units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3.$ Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100.$
A trapezoid has side lengths $3, 5, 7,$ and $11$. The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1} + r_2 \sqrt{n_2} + r_3$, where $r_1, r_2,$ and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to
\[r_1 + r_2 + r_3 + n_1 + n_2?\]
$ \textbf{(A)}\ 57\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65 $
Let a triangle $ABC$ inscribed in circle $c(O,R)$ and $D$ an arbitrary point on $BC$(different from the midpoint).The circumscribed circle of $BOD$,which is $(c_1)$, meets $c(O,R)$ at $K$ and $AB$ at $Z$.The circumscribed circle of $COD$ $(c_2)$,meets $c(O,R)$ at $M$ and $AC$ at $E$.Finally, the circumscribed circle of $AEZ$ $(c_3)$,meets $c(O,R)$ at $N$.Prove that $\triangle{ABC}=\triangle{KMN}.$
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)