Let $\Delta ABC$ be a triangle and $L$ be the foot of the bisector of $\angle A$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle ABL$ and $\triangle ACL$ respectively and let $B_1$ and $C_1$ be the projections of $C$ and $B$ through the bisectors of the angles $\angle B$ and $\angle C$ respectively. The incircle of $\Delta ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively and the bisectors of angles $\angle B$ and $\angle C$ meet the perpendicular bisector of $AL$ at points $Q$ and $P$ respectively. Prove that the five lines $PC_0, QB_0, O_1C_1, O_2B_1$ and $BC$ are all concurrent.

Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[ \angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.

Convex quadrilateral $ABCD$ has right angles $\angle A$ and $\angle C$ and is such that $AB=BC$ and $AD=CD$. The diagonals $AC$ and $BD$ intersect at point $M$. Points $P$ and $Q$ lie on the circumcircle of triangle $AMB$ and segment $CD$, respectively, such that points $P$, $M$, and $Q$ are collinear. Suppose that $m\angle ABC=160^\circ$ and $m\angle QMC=40^\circ$. Find $MP\cdot MQ$, given that $MC=6$.

In the circumscircle of a triangle $ABC$, let $A_1$ be the point diametrically opposed to the vertex $A$. Let $A'$ the intersection point of $AA'$ and $BC$. The perpendicular to the line $AA'$ from $A'$ meets the sides $AB$ and $AC$ at $M$ and $N$, respectively. Prove that the points $A,M,A_1$ and $N$ lie on a circle which has the center on the height from $A$ of the triangle $ABC$.

$11$ people are sitting around a circle table, orderly (means that the distance between two adjacent persons is equal to others) and $11$ cards with numbers $1$ to $11$ are given to them. Some may have no card and some may have more than $1$ card. In each round, one [and only one] can give one of his cards with number $ i $ to his adjacent person if after and before the round, the locations of the cards with numbers $ i-1,i,i+1 $ don’t make an acute-angled triangle. (Card with number $0$ means the card with number $11$ and card with number $12$ means the card with number $1$!) Suppose that the cards are given to the persons regularly clockwise. (Mean that the number of the cards in the clockwise direction is increasing.) Prove that the cards can’t be gathered at one person.

grade IX P4, X P3 The bisectors of the angles $ A $ and $ C $ of the triangle $ ABC $ intersect the circumscirbed circle of this triangle at the points $ A_0 $ and $ C_0 $, respectively. The straight line passing through the center of the inscribed circle of triangle $ ABC $ parallel to the side of $ AC $, intersects with the line $ A_0C_0 $ at $ P $. Prove that the line $ PB $ is tangent to the circumcircle of the triangle $ ABC $. grade XI P4 The bisectors of the angles $ A $ and $ C $ of the triangle $ ABC $ intersect the sides at the points $ A_1 $ and $ C_1 $, and the circumcircle of this triangle at points $ A_0 $ and $ C_0 $ respectively. Straight lines $ A_1C_1 $ and $ A_0C_0 $ intersect at point $ P $. Prove that the segment connecting $ P $ with the center inscribed circles of triangle $ ABC $, parallel to $ AC $.

Let $ABC$ be an acute traingle with $AC \neq BC$. Point $K$ is a foot of altitude through vertex $C$. Point $O$ is a circumcenter of $ABC$. Prove that areas of quadrilaterals $AKOC$ and $BKOC$ are equal.

Let $[ABC]$ be any triangle and let $D, E$ and $F$ be the symmetrics of the circumcenter wrt the three sides. Prove that the triangles $[ABC]$ and $[DEF]$ are congruent. [img]https://cdn.artofproblemsolving.com/attachments/c/6/45bd929dfff87fb8deb09eddb59ef46e0dc0f4.png[/img]

Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.

Let $ABC$ be a triangle such that $AB = AC$. Let $D,E$ be points on $AB,AC$ respectively such that $DE = AC$. Let $DE$ meet the circumcircle of triangle $ABC$ at point $T$. Let $P$ be a point on $AT$. Prove that $PD + PE = AT$ if and only if $P$ lies on the circumcircle of triangle $ADE$.

Distinct points $B,B',C,C'$ lie on an arbitrary line $\ell$. $A$ is a point not lying on $\ell$. A line passing through $B$ and parallel to $AB'$ intersects with $AC$ in $E$ and a line passing through $C$ and parallel to $AC'$ intersects with $AB$ in $F$. Let $X$ be the intersection point of the circumcircles of $\triangle{ABC}$ and $\triangle{AB'C'}$($A \neq X$). Prove that $EF \parallel AX$.

Two circles are tangent to each other internally at a point $\ T $. Let the chord $\ AB $ of the larger circle be tangent to the smaller circle at a point $\ P $. Prove that the line $\ TP $ bisects $\ \angle ATB $.

Let $ABC$ be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to $A$. Suppose that the outer inscribed circle with respect to $A$ touches $BC,AC,AB$ at $M,N,L$. Prove that $O$ (Center of circumcircle) is the orthocenter of $MNL$.

Let $ ABCD$ be a convex quadrilateral. The perpendicular bisectors of its sides $ AB$ and $ CD$ meet at $ Y$. Denote by $ X$ a point inside the quadrilateral $ ABCD$ such that $ \measuredangle ADX \equal{} \measuredangle BCX < 90^{\circ}$ and $ \measuredangle DAX \equal{} \measuredangle CBX < 90^{\circ}$. Show that $ \measuredangle AYB \equal{} 2\cdot\measuredangle ADX$.

Let $ABCD$ be a square and let $\Gamma$ be the circumcircle of $ABCD$. $M$ is a point of $\Gamma$ belonging to the arc $CD$ which doesn't contain $A$. $P$ and $R$ are respectively the intersection points of $(AM)$ with $[BD]$ and $[CD]$, $Q$ and $S$ are respectively the intersection points of $(BM)$ with $[AC]$ and $[DC]$. Prove that $(PS)$ and $(QR)$ are perpendicular.

Diagonals $AC$ and $BD$ of a trapezoid $ABCD$ meet at $P$. The circumcircles of triangles $ABP$ and $CDP$ intersect the line $AD$ for the second time at points $X$ and $Y$ respectively. Let $M$ be the midpoint of segment $XY$. Prove that $BM = CM$.

A triangle has no angle greater than $90^o$. Show that the sum of the medians is greater than four times the circumradius.

We are given triangles $ABC$ and $DEF$ such that $D\in BC, E\in CA, F\in AB$, $AD\perp EF, BE\perp FD, CF\perp DE$. Let the circumcenter of $DEF$ be $O$, and let the circumcircle of $DEF$ intersect $BC,CA,AB$ again at $R,S,T$ respectively. Prove that the perpendiculars to $BC,CA,AB$ through $D,E,F$ respectively intersect at a point $X$, and the lines $AR,BS,CT$ intersect at a point $Y$, such that $O,X,Y$ are collinear. [i]Proposed by Sammy Luo[/i]

Let $D$ be an arbitrary point on side $AB$ of triangle $ABC$. Circumcircles of triangles $BCD$ and $ACD$ intersect sides $AC$ and $BC$ at points $E$ and $F$, respectively. Perpendicular bisector of $EF$ cuts $AB$ at point $M$, and line perpendicular to $AB$ at $D$ at point $N$. Lines $AB$ and $EF$ intersect at point $T$, and the second point of intersection of circumcircle of triangle $CMD$ and line $TC$ is $U$. Prove that $NC=NU$

Let $\Delta ABC$ be a scalene triangle. Points $D,E$ lie on side $\overline{AC}$ in the order, $A,E,D,C$. Let the parallel through $E$ to $BC$ intersect $\odot (ABD)$ at $F$, such that, $E$ and $F$ lie on the same side of $AB$. Let the parallel through $E$ to $AB$ intersect $\odot (BDC)$ at $G$, such that, $E$ and $G$ lie on the same side of $BC$. Prove, Points $D,F,E,G$ are concyclic

Let $ABC$ be a triangle such that $AB < AC$. Let the excircle opposite to A be tangent to the lines $AB, AC$, and $BC$ at points $D, E$, and $F$, respectively, and let $J$ be its centre. Let $P$ be a point on the side $BC$. The circumcircles of the triangles $BDP$ and $CEP$ intersect for the second time at $Q$. Let $R$ be the foot of the perpendicular from $A$ to the line $FJ$. Prove that the points $P, Q$, and $R$ are collinear. (The [i]excircle[/i] of a triangle $ABC$ opposite to $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.) [i]Proposed by Bozhidar Dimitrov, Bulgaria[/i]

Collinear points $A$, $B$, and $C$ are given in the Cartesian plane such that $A= (a, 0)$ lies along the x-axis, $B$ lies along the line $y=x$, $C$ lies along the line $y=2x$, and $\frac{AB}{BC}=2$. If $D= (a, a)$, and the circumcircle of triangle $ADC$ intersects the line $y=x$ again at $E$, and ray $AE$ intersects $y=2x$ at $F$, evaluate $\frac{AE}{EF}$.

Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$. (a) Show that all such lines $ AB$ are concurrent. (b) Find the locus of midpoints of all such segments $ AB$.

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

Let the point $P$ in the interior of the triangle $ABC$. $(AP, (BP, (CP$ intersect the circumcircle of $ABC$ at $A_{1}, B_{1}, C_{1}$. Prove that the maximal value of the sum of the areas $A_{1}BC$, $B_{1}AC$, $C_{1}AB$ is $p(R-r)$, where $p, r, R$ are the usual notations for the triangle $ABC$.