Let $ A$, $ B$ be two fixed points and $ C$ is a variable point on the plane such that $ \angle ACB\equal{}\alpha$ (constant) ($ 0^{\circ}\le \alpha\le 180^{\circ}$). Let $ D$, $ E$, $ F$ be the projections of the incenter $ I$ of triangle $ ABC$ to its sides $ BC$, $ CA$, $ AB$, respectively. Denoted by $ M$, $ N$ the intersections of $ AI$, $ BI$ with $ EF$, respectively. Prove that the length of the segment $ MN$ is constant and the circumcircle of triangle $ DMN$ always passes through a fixed point.

Let $P$ be a point in the interior of quadrilateral $ABCD$ such that: $$\angle BPC=2\angle BAC \ \ ,\ \ \angle PCA = \angle PAD \ \ ,\ \ \angle PDA=\angle PAC$$ Prove that: $$\angle PBD= \left | \angle BCA - \angle PCA \right |$$ [i]Proposed by Ali Zamani[/i]

In an acute triangle $ABC$ points $D,E,F$ are located on the sides $BC,CA, AB$ such that \[ \frac{CD}{CE} = \frac{CA}{CB} , \frac{AE}{AF} = \frac{AB}{AC} , \frac{BF}{FD} = \frac{BC}{BA} \] Prove that $AD,BE,CF$ are altitudes of triangle $ABC$.

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.

Let $\bigtriangleup ABC$ is not an isosceles triangle and is an acute triangle, $AD,BE,CF$ be the altitudes and $H$ is the orthocenter .Let $I$ is the circumcenter of $\bigtriangleup HEF$ and let $K,J$ is the midpoint of $BC,EF$ respectively.Let $HJ$ intersects $(I)$ again at $G$ and $GK$ intersects $(I)$ at $L\neq G$. a) Prove that $AL$ is perpendicular to $EF$. b) Let $AL$ intersects $EF$ at $M$, the line $IM$ intersects the circumcircle $\bigtriangleup IEF$ again at $N$, $DN$ intersects $AB,AC$ at $P$ and $Q$ respectively then prove that $PE,QF,AK$ are concurrent.

In the rhombus $ABCD,$ the measure of $\angle{B}=40^{\circ}, E$ is the midpoint of $BC,$ and $F$ is the base of the perpendicular dropped from $A$ on $DE.$ Find the measure of $\angle{DFC}.$

Let $BD$ be the external bisector of a triangle $ABC$ with $AB > BC$; $K$ and $K_1$ be the touching points of side $AC$ with the incircle and the excircle centered at $I$ and $I_1$ respectively. The lines $BK$ and $DI_1$ meet at point $X$, and the lines $BK_1$ and $DI$ meet at point $Y$. Prove that $XY \perp AC$.

Given a triangle $ \triangle{ABC} $. Denote its incircle and circumcircle by $ \omega, \Omega $, respectively. Assume that $ \omega $ tangents the sides $ AB, AC $ at $ F, E $, respectively. Then, let the intersections of line $ EF $ and $ \Omega $ to be $ P,Q $. Let $ M $ to be the mid-point of $ BC $. Take a point $ R $ on the circumcircle of $ \triangle{MPQ} $, say $ \Gamma $, such that $ MR \perp EF $. Prove that the line $ AR $, $ \omega $ and $ \Gamma $ intersect at one point.

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $P$ and $Q$ be points in the half plane defined by $BC$ containing $A$, such that $BP$ and $CQ$ are tangents to $\Gamma$ and $PB = BC = CQ$. Let $K$ and $L$ be points on the external bisector of the angle $\angle CAB$ , such that $BK = BA, CL = CA$. Let $M$ be the intersection point of the lines $PK$ and $QL$. Prove that $MK=ML$.

Let $ABC$ be a triangle with midpoint $M$ of $BC$. A point $X$ is called [i]immaculate[/i] if the perpendicular line from $X$ to line $MX$ intersects lines $AB$ and $AC$ at two points that are equidistant from $M$. Suppose $U, V, W$ are three immaculate points on the circumcircle of triangle $ABC$. Prove that $M$ is the incentre of $\triangle UVW$. [i]Proposed by Pranjal Srivastava and Rohan Goyal[/i]

Let $D$ and $E$ be points in the interiors of sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$. Let the lines $CD$ and $BE$ meet at $F$. Prove that the incentre $I$ of triangle $ABC$, the orthocentre $H$ of triangle $DEF$ and the midpoint $M$ of the arc $BAC$ of the circumcircle of triangle $ABC$ are collinear.

Let $ABCD$ be an circumscribed quadrilateral such that $m(\widehat{A})=m(\widehat{B})=120^\circ$, $m(\widehat{C})=30^\circ$, and $|BC|=2$. What is $|AD|$? $ \textbf{(A)}\ \sqrt 3 - 1 \qquad\textbf{(B)}\ \sqrt 2 - 3 \qquad\textbf{(C)}\ \sqrt 6 - \sqrt 2 \qquad\textbf{(D)}\ 2 - \sqrt 2 \qquad\textbf{(E)}\ 3 - \sqrt 3 $

Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon. Find the area of this hexagon. [asy] import geometry; size(10cm); point O1=(0,0),O2=(15,0),B=9*dir(30); circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B); point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0]; filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black); draw(w1); draw(w2); draw(O1--O2,dashed); draw(o); dot(O1); dot(O2); dot(A); dot(D); dot(C); dot(B); label("$\omega_1$",8*dir(110),SW); label("$\omega_2$",5*dir(70)+(15,0),SE); label("$O_1$",O1,W); label("$O_2$",O2,E); label("$B$",B,N+1/2*E); label("$A$",A,N+1/2*W); label("$C$",C,S+1/4*W); label("$D$",D,S+1/4*E); label("$15$",midpoint(O1--O2),N); label("$16$",midpoint(C--D),N); label("$2$",midpoint(A--B),S); label("$\Omega$",o.C+(o.r-1)*dir(270)); [/asy]

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]