Found problems: 847
The point $M$ is the middle of the side $BC$ of the acute-angled triangle $ABC$ and the points $E$ and $F$ are respectively perpendicular foot of $M$ to the sides $AC$ and $AB$. The points $X$ and $Y$ lie on the plane such that $\triangle XEC\sim\triangle CEY$ and $\triangle BYF\sim\triangle XBF$(The vertices of triangles with this order are corresponded in the similarities) and the points $E$ and $F$ [u]don't[/u][neither] lie on the line $XY$. Prove that $XY\perp AM$.
In an acute triangle $ABC$ with $AC<BC$, lines $m_a$ and $m_b$ are the perpendicular bisectors of sides $BC$ and
$AC$, respectively. Further, let $M_c$ be the midpoint of side $AB$. The Median $CM_c$ intersects $m_a$ in point $S_a$ and $m_b$ in point $S_b$; the lines $AS_b$ und $BS_a$ intersect in point $K$.
Prove: $\angle ACM_c = \angle KCB$.
In triangle $ABC$, the interior and exterior angle bisectors of $ \angle BAC$ intersect the line $BC$ in $D $ and $E$, respectively. Let $F$ be the second point of intersection of the line $AD$ with the circumcircle of the triangle $ ABC$. Let $O$ be the circumcentre of the triangle $ ABC $and let $D'$ be the reflection of $D$ in $O$. Prove that $ \angle D'FE =90.$
Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
Let $ABC$ be a triangle. Let $I$ be the Incenter of $ABC$ and $S$ be the midpoint of arc $BAC$. Define $IA$ as the $A$-excenter wrt $ABC$. Define $\omega$ to be the circle centred at $S$ with radius $SB$. Let $AI_A \cap \omega = X$, $Y$. Show that $\angle BCX = \angle ACY$.
Let $ABC$ be an non-isosceles triangle with incenter $I$, circumcenter $O$ and a point $D$ on segment $BC $such that $(BID) $cut segments $AB $ at$ E $and $(CID) $cuts segment $AC $at $F$ Circle $(DEF)$ cuts segments $AB$,$AC $again at $M,N$. Let $P$ The intersection of $IB$ and $DE $ , $Q$ The intersection of $IC$and $DF$ . Prove that $EN,FM,PQ $are parallel and the median of vertex $I$in triangle $IPQ$ bisects the arc $BAC$ of $(O)$.
Let $ABC$ be an acute-angled triangle. Point $P$ is such that $AP = AB$ and $PB \parallel AC$. Point $Q$ is such that $AQ = AC$ and $CQ \parallel AB$. Segments $CP$ and $BQ$ meet at point $X$. Prove that the circumcenter of triangle $ABC$ lies on the circumcircle of triangle $PXQ$.
Let $ABC$ be an acute-angled triangle and let $P$ be the intersection of the tangents at $B$ and $C$ of the circumscribed circle of $\vartriangle ABC$. The line through $A$ perpendicular on $AB$ and cuts the line perpendicular on $AC$ through $C$ at $X$. The line through $A$ perpendicular on $AC$ cuts the line perpendicular on $AB$ through $B$ at $Y$. Show that $AP \perp XY$.
Let $ABC$ be a triangle with side lengths $AB=6, BC=7, CA=8$ and circumcircle $\omega.$ Denote $M$ to be the midpoint of $BC.$ Let $P$ be the intersection of the tangent to $\omega$ at $A$ and $BC.$ The line parallel to $BC$ passing through $A$ intersects $\omega$ at another point $D.$ The tangent to $\omega$ passing through $P$ that is not $PA$ intersects $DM$ at a point $Q.$ Denote $J$ to be the intersection of $(BMQ)$ and $AQ.$ Extend $BJ$ to intersect $AC$ at $E.$ Compute $\tfrac{BJ}{JE}.$
[i]Lightning 4.4[/i]
Let $ABC$ be an acute triangle and $\Gamma$ be its circumcircle. Line $\ell$ is tangent to $\Gamma$ at $A$ and let $D$ and $E$ be distinct points on $\ell$ such that $AD = AE$. Suppose that $B$ and $D$ lie on the same side of line $AC$. The circumcircle $\Omega_1$ of $\vartriangle ABD$ meets $AC$ again at $F$. The circumcircle $\Omega_2$ of $\vartriangle ACE$ meets $AB$ again at $G$. The common chord of $\Omega_1$ and $\Omega_2$ meets $\Gamma$ again at $H$. Let $K$ be the reflection of $H$ across line $BC$ and let $L$ be the intersection of $BF$ and $CG$. Prove that $A, K$ and $L$ are collinear.
Let $ABC$ be an acute-angled triangle such that $|AB|<|AC|$. Let $X$ and $Y$ be points on the minor arc ${BC}$ of the circumcircle of $ABC$ such that $|BX|=|XY|=|YC|$. Suppose that there exists a point $N$ on the segment $\overline{AY}$ such that $|AB|=|AN|=|NC|$. Prove that the line $NC$ passes through the midpoint of the segment $\overline{AX}$. \\ \\ (Ivan Novak)
Let $ABC$ be a triangle. $I$ is the incenter, and the incircle is tangent to $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. $P$ is the second point of intersection of $AD$ and the incircle. If $M$ is the midpoint of $EF$, show that $P$, $I$, $M$, $D$ are concyclic.
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 $\angle BAC=90^{\circ}$.
Given that $ABC$ is a triangle, points $A_i, B_i, C_i \hspace{0.15cm} (i \in \{1,2,3\})$ and $O_A, O_B, O_C$ satisfy the following criteria:
a) $ABB_1A_2, BCC_1B_2, CAA_1C_2$ are rectangles not containing any interior points of the triangle $ABC$,
b) $\displaystyle \frac{AB}{BB_1} = \frac{BC}{CC_1} = \frac{CA}{AA_1}$,
c) $AA_1A_3A_2, BB_1B_3B_2, CC_1C_3C_2$ are parallelograms, and
d) $O_A$ is the centroid of rectangle $BCC_1B_2$, $O_B$ is the centroid of rectangle $CAA_1C_2$, and $O_C$ is the centroid of rectangle $ABB_1A_2$.
Prove that $A_3O_A, B_3O_B,$ and $C_3O_C$ concur at a point.
[i]Proposed by Farras Mohammad Hibban Faddila[/i]
Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.
[i]Proposed by El Salvador[/i]
In a non-isosceles triangle $ABC$, let $I$ be its incenter. The incircle of $ABC$ touches $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. A line passing through $D$ and perpendicular to $AD$ intersects $IB$ and $IC$ at $A_b$ and $A_c$, respectively. Define the points $B_c$, $B_a$, $C_a$, and $C_b$ similarly. Let $G$ be the intersection of the cevians $AD$, $BE$, and $CF$. The points $O_1$ and $O_2$ are the circumcenter of the triangles $A_bB_cC_a$ and $A_cB_aC_b$, respectively. Prove that $IG$ is the perpendicular bisector of $O_1O_2$.
Let $\omega$ be a circle with center $O$ and let $A$ be a point outside $\omega$. The tangents from $A$ touch $\omega$ at points $B$, and $C$. Let $D$ be the point at which the line $AO$ intersects the circle such that $O$ is between $A$ and $D$. Denote by $X$ the orthogonal projection of $B$ onto $CD$, by $Y$ the midpoint of the segment $BX$ and by $Z$ the second point of intersection of the line $DY$ with $\omega$. Prove that $ZA$ and $ZC$ are perpendicular to each other.
Let $ABC$ be a non-right triangle and let $M$ be the midpoint of $BC$. Let $D$ be a point on $AM$ (D≠A, D≠M). Let ω1 be a circle through $D$ that intersects $BC$ at $B$ and let ω2 be a circle through $D$ that intersects $BC$ at $C$. Let $AB$ intersect ω1 at $B$ and $E$, and let $AC$ intersect ω2 at $C$ and $F$.
Prove, that the tangent on ω1 at $E$ and the tangent on ω2 at $F$ intersect on $AM$.
A line through $A$ intersects a circle at points $B,C$ with $B$ between $A,C$. The two tangents from $A$ intersect the circle at $S,T$. $ST$ and $AC$ intersect at $P$. Show that $\frac{AP}{PC}=2\frac{AB}{BC}$.
Let $C$ and $D$ be two points on the semicricle with diameter $AB$ such that $B$ and $C$ are on distinct sides of the line $AD$. Denote by $M$, $N$ and $P$ the midpoints of $AC$, $BD$ and $CD$ respectively. Let $O_A$ and $O_B$ the circumcentres of the triangles $ACP$ and $BDP$. Show that the lines $O_AO_B$ and $MN$ are parallel.
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$, and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$. The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$. Let $E$ be the second point of intersection between $AQ$ and $\omega$. Prove that $B$, $E$, $R$ are collinear.
In a triangle $ABC$, let $BD$ be its altitude and let $H$ be its orthocenter. The perpendicular bisector of of $HD$ meets $(BCD)$ at $P, Q$. Prove that $\angle APB+\angle AQB=180^{o}$
Let $ABCD$ be a trapezium with $AD||BC$; and let $X$ and $Y$ be the midpoints of $AC$ and $BD$ respectively. Prove that if $\angle DAY=\angle CAB$ then the internal bisectors of $\angle XAY$ and $\angle XBY$ meet on $XY$.
Proposed by Belur Jana Venkatachala
$ABC$ is an acute triangle with $AB> AC$. $\Gamma_B$ is a circle that passes through $A,B$ and is tangent to $AC$ on $A$. Define similar for $ \Gamma_C$. Let $D$ be the intersection $\Gamma_B$ and $\Gamma_C$ and $M$ be the midpoint of $BC$. $AM$ cuts $\Gamma_C$ at $E$. Let $O$ be the center of the circumscibed circle of the triangle $ABC$. Prove that the circumscibed circle of the triangle $ODE$ is tangent to $\Gamma_B$.