Let $ I(A)$ and $ I(B)$ be the centers of the excircles of a triangle $ ABC,$ which touches the sides $ BC$ and $ CA$ in its interior. Furthermore let $ P$ a point on the circumcircle $ \omega$ of the triangle $ ABC.$ Show that the center of the segment which connects the circumcenters of the triangles $ I(A)CP$ and $ I(B)CP$ coincides with the center of the circle $ \omega.$

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 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]

Let $MAZN$ be an isosceles trapezium inscribed in a circle $(c)$ with centre $O$. Assume that $MN$ is a diameter of $(c)$ and let $ B$ be the midpoint of $AZ$. Let $(\epsilon)$ be the perpendicular line on $AZ$ passing through $ A$. Let $C$ be a point on $(\epsilon)$, let $E$ be the point of intersection of $CB$ with $(c)$ and assume that $AE$ is perpendicular to $CB$. Let $D$ be the point of intersection of $CZ$ with $(c)$ and let $F$ be the antidiametric point of $D$ on $(c)$. Let $ P$ be the point of intersection of $FE$ and $CZ$. Assume that the tangents of $(c)$ at the points $M$ and $Z$ meet the lines $AZ$ and $PA$ at the points $K$ and $T$ respectively. Prove that $OK$ is perpendicular to $TM$. Theoklitos Parayiou, Cyprus

Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.

Let $ABC$ be a triangle such that $\angle ACB=2\angle ABC$. Let $D$ be the point on the side $BC$ such that $CD=2BD$. The segment $AD$ is extended to $E$ so that $AD=DE$. Prove that \[ \angle ECB+180^{\circ }=2\angle EBC. \]

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.

Let $ ABC$ be a tringle and let $ P$ be an interior point such that $ \angle BPC \equal{} 90 ,\angle BAP \equal{} \angle BCP$.Let $ M,N$ be the mid points of $ AC,BC$ respectively.Suppose $ BP \equal{} 2PM$.Prove that $ A,P,N$ are collinear.

$A,B,C$ are on circle $\mathcal C$. $I$ is incenter of $ABC$ , $D$ is midpoint of arc $BAC$. $W$ is a circle that is tangent to $AB$ and $AC$ and tangent to $\mathcal C$ at $P$. ($W$ is in $\mathcal C$) Prove that $P$ and $I$ and $D$ are on a line.

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?

The length of the altitude of equilateral triangle $ ABC$ is $3$. A circle with radius $2$, which is tangent to $ \left[BC\right]$ at its midpoint, meets other two sides. If the circle meets $ AB$ and $ AC$ at $ D$ and $ E$, at the outer of $\triangle ABC$ , find the ratio $ \frac {Area\, \left(ABC\right)}{Area\, \left(ADE\right)}$. $\textbf{(A)}\ 2\left(5 \plus{} \sqrt {3} \right) \qquad\textbf{(B)}\ 7\sqrt {2} \qquad\textbf{(C)}\ 5\sqrt {3} \\ \qquad\textbf{(D)}\ 2\left(3 \plus{} \sqrt {5} \right) \qquad\textbf{(E)}\ 2\left(\sqrt {3} \plus{} \sqrt {5} \right)$

In a triangle $\triangle ABC$ with$\angle B>\angle C$, the altitude, the angle bisector, and the median from $A$ intersect $BC$ at $H, L$ and $D$, respectively. Show that $\angle HAL=\angle DAL$ if and only if $\angle BAC=90^{\circ}$.

Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

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 $K$ be the measure of the area bounded by the $x$-axis, the line $x=8$, and the curve defined by \[f=\{(x,y)\,|\, y=x\text{ when }0\leq x\leq 5,\,y=2x-5\text{ when }5\leq x\leq 8\}.\] Then $K$ is: $\textbf{(A) }21.5\qquad \textbf{(B) }36.4\qquad \textbf{(C) }36.5\qquad \textbf{(D) }44\qquad$ $\textbf{ (E) }\text{less than 44 but arbitrarily close to it.}$

Let $ABCD$ be an isosceles trapezoid with $AD\parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP\perp CD$ and $BQ\perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX=3$ and $XQ=1$. Compute the largest possible area of $ABCD$.

Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$. [i]Proposed by Charles Leytem, Luxembourg[/i]

Let $ABCD$ be a trapezoid ($AD//BC$) with $\angle A=\angle B= 90^o$ and $AD<BC$. Let $E$ be the intersection point of the non parallel sides $AB$ and $CD$, $Z$ be the symmetric point of $A$ wrt line $BC$ and $M$ be the midpoint of $EZ$. If it is given than line $CM$ is perpendicular on line $DZ$, then prove that line $ZC$ is perpendicular on line $EC$.

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$.)

Consider the areas of the four triangles obtained by drawing the diagonals $AC$ and $BD$ of a trapezium $ABCD$. The product of these areas, taken two at time, are computed. If among the six products so obtained, two products are 1296 and 576, determine the square root of the maximum possible area of the trapezium to the nearest integer.

Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change? $\mathrm{(A)}\ \text{the length of the segment} MN$ $\mathrm{(B)}\ \text{the perimeter of }\triangle PAB$ $\mathrm{(C)}\ \text{ the area of }\triangle PAB$ $\mathrm{(D)}\ \text{ the area of trapezoid} ABNM$ [asy] draw((2,0)--(8,0)--(6,4)--cycle); draw((4,2)--(7,2)); draw((1,4)--(9,4),Arrows); label("$A$",(2,0),SW); label("$B$",(8,0),SE); label("$M$",(4,2),W); label("$N$",(7,2),E); label("$P$",(6,4),N);[/asy] $\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 1 \qquad\mathrm{(C)}\ 2 \qquad\mathrm{(D)}\ 3 \qquad\mathrm{(E)}\ 4$

Let $ABCD$ be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at $E$. The midpoints of $AB$ and $CD$ are $F$ and $G$ respectively, and $\ell$ is the line through $G$ parallel to $AB$. The feet of the perpendiculars from E onto the lines $\ell$ and $CD$ are $H$ and $K$, respectively. Prove that the lines $EF$ and $HK$ are perpendicular.

Let $ABCD$ be an isosceles trapezoid ($AD\parallel BC$), $\angle BAD = 80^o$, $\angle BDA = 60^o$. Point $P$ lies on $CD$ and $\angle PAD = 50^o$. Find $\angle PBC$

Let $ABCD$ be a trapezoid which is not a parallelogram, such that $AD$ is parallel to $BC$. Let $O=BD\cap AC$ and $S$ be the second intersection of the circumcircles of triangles $AOB$ and $DOC$. Prove that the circumcircles of triangles $ASD$ and $BSC$ are tangent.

An isosceles trapezoid has legs and shorter base of length $1$. Find the maximum possible value of its area