Let $ABC$ be a triangle, $H_a, H_b$ and $H_c$ the feet of its altitudes from $A, B$ and $C$, respectively, $T_a, T_b, T_c$ its touchpoints of the incircle with the sides $BC, CA$ and $AB$, respectively. The circumcircles of triangles $AH_bH_c$ and $AT_bT_c$ intersect again at $A'$. The circumcircles of triangles $BH_cH_a$ and $BT_cT_a$ intersect again at $B'$. The circumcircles of triangles $CH_aH_b$ and $CT_aT_b$ intersect again at $C'$. Prove that the points $A',B',C'$ are collinear. Malik Talbi

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] $\textbf {(A) } 30 \qquad \textbf {(B) } 33 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 42$

Let $\vartriangle ABC$ be an acute triangle with incenter $I$. The line passing through $I$ parallel to $AC$ intersects $AB$ at $M$, and the line passing through $I$ parallel to $AB$ intersects $AC$ at $N$. Let the line $MN$ intersect the circumcircle of $\vartriangle ABC$ at $X$ and $Y$ . Let $Z$ be the midpoint of arc $BC$ (not containing $A$). Prove that $I$ is the orthocenter of $\vartriangle XY Z$

Let $ABC$ be an acute angled triangle with incenter $I$. Line perpendicular to $BI$ at $I$ meets $BA$ and $BC$ at points $P$ and $Q$ respectively. Let $D, E$ be the incenters of $\triangle BIA$ and $\triangle BIC$ respectively. Suppose $D,P,Q,E$ lie on a circle. Prove that $AB=BC$.

Let $ABC$ be a triangle such that $AB > BC$ and let $D$ be a variable point on the line segment $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$, lying on the opposite side of $BC$ from $A$ such that $\angle BAE = \angle DAC$. Let $I$ be the incenter of triangle $ABD$ and let $J$ be the incenter of triangle $ACE$. Prove that the line $IJ$ passes through a fixed point, that is independent of $D$. [i]Proposed by Merlijn Staps[/i]

Let $AA_1, BB_1, $ and $CC_1$ be the bisectors of a triangle $ABC$. The segments $BB_1$ and $A_1C_1$ meet at point $D$. Let $E$ be the projection of $D$ to $AC$. Points $P$ and $Q$ on sides $AB$ and $BC$ respectively are such that $EP = PD, EQ = QD$. Prove that $\angle PDB_1 = \angle EDQ$.

Let $ ABC$ be a triangle with $ \angle A = 60^{\circ}$.Prove that if $ T$ is point of contact of Incircle And Nine-Point Circle, Then $ AT = r$, $ r$ being inradius.

In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that \[ \frac {E}{E_1} \geq 2. \]

Let $ABC$ be an acute triangle with $AC > AB$. Let $\Gamma$ be the circle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smaller arc $BC$ of this circle. Let $I$ be the incenter of $ABC$ and let $E$ and $F$ be points on sides $AB$ and $AC$, respectively, such that $AE = AF$ and $I$ lies on the segment $EF$. Let $P$ be the second intersection point of the circumcircle of the triangle $AEF$ with $\Gamma$ with $P \ne A$. Let $G$ and $H$ be the intersection points of the lines $PE$ and $PF$ with $\Gamma$ different from $P$, respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines AB and $AC$, respectively. Show that the line $JK$ passes through the midpoint of $BC$.

In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that: \[AL+BM+CN \leq 3(AD+BE+CF)\] When does equality occur?

In an acute-angled triangle $ABC$, the orthocenter is $H$. $I_H$ is the incenter of $\vartriangle BHC$. The bisector of $\angle BAC$ intersects the perpendicular from $I_H$ to the side $BC$ at point $K$. Let $F$ be the foot of the perpendicular from $K$ to $AB$. Prove that $2KF+BC=BH +HC$ A. Voidelevich

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 $AR\parallel BC$.

Let $ABC$ be an acute angled triangle with $\angle{BAC}=60^\circ$ and $AB > AC$. Let $I$ be the incenter, and $H$ the orthocenter of the triangle $ABC$ . Prove that $2\angle{AHI}= 3\angle{ABC}$.

Triangle $ ABC$ has incentre $ I$ and the incircle touches $ BC, CA$ at $ D, E$ respectively. Let $ BI$ meet $ DE$ at $ G$. Show that $ AG$ is perpendicular to $ BG$.

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

Let $H$ be the orthocenter of the triangle $ABC$ and let $D$, $E$, $F$ be the feet of heights by $A$, $B$, $C$. Let $\omega_D$, $\omega_E$, $\omega_F$ be the incircles of $FEH$, $DHF$, $HED$ and let $I_D$, $I_E$, $I_F$ be their centers. Show that $I_DD$, $I_EE$ and $I_FF$ compete.

Let $I$ be the incentre of triangle $ABC$ with $AB>AC$ and let the line $AI$ intersect the side $BC$ at $D$. Suppose that point $P$ lies on the segment $BC$ and satisfies $PI=PD$. Further, let $J$ be the point obtained by reflecting $I$ over the perpendicular bisector of $BC$, and let $Q$ be the other intersection of the circumcircles of the triangles $ABC$ and $APD$. Prove that $\angle BAQ=\angle CAJ$.

Let $ ABC$ be a triangle, and $ I$ its incenter. Let the incircle of $ ABC$ touch side $ BC$ at $ D$, and let lines $ BI$ and $ CI$ meet the circle with diameter $ AI$ at points $ P$ and $ Q$, respectively. Given $ BI \equal{} 6, CI \equal{} 5, DI \equal{} 3$, determine the value of $ \left( DP / DQ \right)^2$.

Let scalene triangle $ABC$ have circumcenter $O$ and incenter $I$. Its incircle $\omega$ is tangent to sides $BC,CA,$ and $AB$ at $D,E,$ and $F$, respectively. Let $P$ be the foot of the altitude from $D$ to $EF$, and let line $DP$ intersect $\omega$ again at $Q \ne D$. The line $OI$ intersects the altitude from $A$ to$ BC$ at $T$. Given that $OI \|BC,$ show that $PQ=PT$.

The incircle of triangle $ABC$ is tangent to the sides $AB,AC$ and $BC$ at the points $M,N$ and $K$ respectively. The median $AD$ of the triangle $ABC$ intersects $MN$ at the point $L$. Prove that $K,I$ and $L$ are collinear, where $I$ is the incenter of the triangle $ABC$.

Let $AA _1$ and $CC_1$ be altitudes of an acute triangle $ABC$. Let $I$ and $J$ be the incenters of the triangles $AA_1C$ and $AC_1C$ respectively. The $C_1J$ and $A_1 I$ lines cut into $T$. Prove that lines $AT$ and $TC$ are perpendicular.

The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$. Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF.$

Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.

The triangle $ABC$ is given. Let $A'$ be the midpoint of the side $BC$, $B_c{}$ be the projection of $B{}$ onto the bisector of the angle $ACB{}$ and $C_b$ be the projection of the point $C{}$ onto the bisector of the angle $ABC$. Let $A_0$ be the center of the circle passing through $A', B_c, C_b$. The points $B_0$ and $C_0$ are defined similarly. Prove that the incenter of the triangle $ABC$ coincides with the orthocenter of the triangle $A_0B_0C_0$.

Let $P_1, P_2, P_3, P_4$ be four points on a circle, let $I_1$ be incenter of the triangle of vertices $P_2P_3P_4$, $I_2$ the incenter of the triangle $P_1P_3P_4$, $I_3$ the incenter of the triangle $P_1P_2P_4$, $I_4$ the incenter of the triangle $P_2P_3P_1$. Prove that $I_1I_2I_3I_4$ is a rectangle.