Consider a circle $S$, and a point $P$ outside it. The tangent lines from $P$ meet $S$ at $A$ and $B$, respectively. Let $M$ be the midpoint of $AB$. The perpendicular bisector of $AM$ meets $S$ in a point $C$ lying inside the triangle $ABP$. $AC$ intersects $PM$ at $G$, and $PM$ meets $S$ in a point $D$ lying outside the triangle $ABP$. If $BD$ is parallel to $AC$, show that $G$ is the centroid of the triangle $ABP$. [i]Arnoldo Aguilar (El Salvador)[/i]

Let $ABC$ be a triangle such that $AB<AC$. The perpendicular bisector of the side $BC$ meets the side $AC$ at the point $D$, and the (interior) bisectrix of the angle $ADB$ meets the circumcircle $ABC$ at the point $E$. Prove that the (interior) bisectrix of the angle $AEB$ and the line through the incentres of the triangles $ADE$ and $BDE$ are perpendicular.

Let $D$ be the point on the side $BC$ of the triangle $ABC$ such that $AD$ is the bisector of $\angle CAB$. Let $I$ be the incenter of$ ABC.$ [i](a)[/i] Construct the points $P$ and $Q$ on the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$ and the perimeter of the triangle $APQ$ is equal to $k \cdot BC$, where $k$ is a given rational number. [i](b) [/i]Let $R$ be the intersection point of $PQ$ and $AD$. For what value of $k$ does the equality $AR = RI$ hold? [i](c)[/i] In which case do the equalities $AR = RI = ID$ hold?

Let $I$ be an incenter of $\triangle ABC$. Denote $D, \ S \neq A$ intersections of $AI$ with $BC, \ O(ABC)$ respectively. Let $K, \ L$ be incenters of $\triangle DSB, \ \triangle DCS$. Let $P$ be a reflection of $I$ with the respect to $KL$. Prove that $BP \perp CP$.

$I$ is the incenter of triangle $ABC$. perpendicular from $I$ to $AI$ meet $AB$ and $AC$ at ${B}'$ and ${C}'$ respectively . Suppose that ${B}''$ and ${C}''$ are points on half-line $BC$ and $CB$ such that $B{B}''=BA$ and $C{C}''=CA$. Suppose that the second intersection of circumcircles of $A{B}'{B}''$ and $A{C}'{C}''$ is $T$. Prove that the circumcenter of $AIT$ is on the $BC$.

A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.

Given a triangle $ABC$ with incenter $I$ . It is known that $E_A$ is center of the ex-circle tangent to $BC$. Likewise, $E_B$ and $E_C$ are the centers of the ex-circles tangent to $AC$ and $AB$, respectively. Prove that $I$ is the orthocenter of the triangle $E_AE_BE_C$.

The incircle of a triangle $ABC$ touches the sides $BC,CA,AB$ at points $D,E,F$, respectively. Let $X$ be a point on the incircle, different from the points $D,E,F$. The lines $XD$ and $EF,XE$ and $FD,XF$ and $DE$ meet at points $J,K,L$, respectively. Let further $M,N,P$ be points on the sides $BC,CA,AB$, respectively, such that the lines $AM,BN,CP$ are concurrent. Prove that the lines $JM,KN$ and $LP$ are concurrent. [i]Dinu Serbanescu[/i]

Circle $\omega$ is inscribed in the $\vartriangle ABC$, with center $I$. Using only a ruler, divide segment $AI$ in half. (Grigory Filippovsky)

In a scalene triangle $\Delta ABC$, $AB<AC$, $PB$ and $PC$ are tangents of the circumcircle $(O)$ of $\Delta ABC$. A point $R$ lies on the arc $\widehat{AC}$(not containing $B$), $PR$ intersects $(O)$ again at $Q$. Suppose $I$ is the incenter of $\Delta ABC$, $ID \perp BC$ at $D$, $QD$ intersects $(O)$ again at $G$. A line passing through $I$ and perpendicular to $AI$ intersects $AB,AC$ at $M,N$, respectively. Prove that, if $AR \parallel BC$, then $A,G,M,N$ are concyclic.

Let $ABC$ be a scalene (i.e. non-isosceles) triangle. Let $U$ be the center of the circumcircle of this triangle and $I$ the center of the incircle. Assume that the second point of intersection different from $C$ of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$ lies on the perpendicular bisector of $UI$. Show that $\gamma$ is the second-largest angle in the triangle $ABC$.

In an acute-angled triangle $ABC, \angle ABC$ is the largest angle. The perpendicular bisectors of $BC$ and $BA$ intersect AC at $X$ and $Y$ respectively. Prove that circumcentre of triangle $ABC$ is incentre of triangle $BXY$ .

Let $ABC$ be a triangle with incenter $I$, and let the tangency points of its incircle with its sides $AB$, $BC$, $CA$ be $C'$, $A'$ and $B'$ respectively. Prove that the circumcenters of $AIA'$, $BIB'$, and $CIC'$ are collinear.

Let $ I$ be the incentre and $ O$ the circumcentre of a given acute triangle $ ABC$. The incircle is tangent to $ BC$ at $ D$. Assume that $ \angle B < \angle C$ and the segments $ AO$ and $ HD$ are parallel, where $H$ is the orthocentre of triangle $ABC$. Let the intersection of the line $ OD$ and $ AH$ be $ E$. If the midpoint of $ CI$ is $ F$, prove that $ E,F,I,O$ are concyclic.

Let $ABCD$ be a cyclic quadrilateral, and angle bisectors of $\angle BAD$ and $\angle BCD$ meet at point $I$. Show that if $\angle BIC = \angle IDC$, then $I$ is the incenter of triangle $ABD$.

Let $ABC$ be a triangle with incenter $I$. The line $BI$ meets $AC$ at $D$. Let $P$ be a point on $CI$ such that $DI=DP$ ($P\ne I$), $E$ the second intersection point of segment $BC$ with the circumcircle of $ABD$ and $Q$ the second intersection point of line $EP$ with the circumcircle of $AEC$. Prove that $\angle PDQ=90^\circ$. [i]Proposed by Ariel García[/i]

Given $\triangle ABC$ inscribed in $(O)$ an let $I$ and $I_a$ be it's incenter and $A$-excenter ,respectively. Tangent lines to $(O)$ at $C,B$ intersect the angle bisector of $A$ at $M,N$ ,respectively. Second tangent lines through $M,N$ intersect $(O)$ at $X,Y$. Prove that $XYII_a$ is cyclic.

Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.

In the adjoining diagram, $BO$ bisects $\angle CBA$, $CO$ bisects $\angle ACB$, and $MN$ is parallel to $BC$. If $AB=12$, $BC=24$, and $AC=18$, then the perimeter of $\triangle AMN$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, C=(24,0), A=intersectionpoints(Circle(B,12), Circle(C,18))[0], O=incenter(A,B,C), M=intersectionpoint(A--B, O--O+40*dir(180)), N=intersectionpoint(A--C, O--O+40*dir(0)); draw(B--M--O--B--C--O--N--C^^N--A--M); label("$A$", A, dir(90)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); label("$M$", M, dir(90)*dir(B--A)); label("$N$", N, dir(90)*dir(A--C)); label("$O$", O, dir(90));[/asy] $\textbf {(A) } 30 \qquad \textbf {(B) } 33 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 42$

The interior bisector of $\angle A$ from $\triangle ABC$ intersects the side $BC$ and the circumcircle of $\Delta ABC$ at $D,M$, respectively. Let $\omega$ be a circle with center $M$ and radius $MB$. A line passing through $D$, intersects $\omega$ at $X,Y$. Prove that $AD$ bisects $\angle XAY$.

Point $X$ moves along side $AB$ of triangle $ABC$, and point $Y$ moves along its circumcircle in such a way that line $XY$ passes through the midpoint of arc $AB$. Find the locus of the circumcenters of triangles $IXY$ , where I is the incenter of $ ABC$.

Bisectors $AA_1$, $BB_1$, and $CC_1$ of triangle $ABC$ meet at point $I$. The perpendicular bisector to $BB_1$ meets $AA_1,CC_1$ at points $A_0,C_0$ respectively. Prove that the circumcircles of triangles $A_0IC_0$ and $ABC$ touch.

In triangle $ABC$, let the circumcenter, incenter, and orthocenter be $O$, $I$, and $H$ respectively. Segments $AO$, $AI$, and $AH$ intersect the circumcircle of triangle $ABC$ at $D$, $E$, and $F$. $CD$ intersects $AE$ at $M$ and $CE$ intersects $AF$ at $N$. Prove that $MN$ is parallel to $BC$.

Let $ABC$ be a triangle with incenter $I$, and $A$-excenter $\Gamma$. Let $A_1,B_1,C_1$ be the points of tangency of $\Gamma$ with $BC,AC$ and $AB$, respectively. Suppose $IA_1, IB_1$ and $IC_1$ intersect $\Gamma$ for the second time at points $A_2,B_2,C_2$, respectively. $M$ is the midpoint of segment $AA_1$. If the intersection of $A_1B_1$ and $A_2B_2$ is $X$, and the intersection of $A_1C_1$ and $A_2C_2$ is $Y$, prove that $MX=MY$.

Let $I$ be the incenter of the acute triangle $\triangle ABC$, and $BI$, $CI$ intersect the altitude of $\triangle ABC$ through $A$ at $U$, $V$, respectively. The circle with $AI$ as a diameter intersects $\odot(ABC)$ again at $T$, and $\odot(TUV)$ intersects the segment $BC$ and $\odot(ABC)$ at $P$, $Q$, respectively. Let $R$ be another intersection of $PQ$ and $\odot(ABC)$. Show that $AR\parallel BC$.