Found problems: 3585
In an acute-angled triangle $ABC$, the altitudes intersect at point $H$, and point $K$ is the foot of the altitude drawn from the vertex $A$. Circle $c$ passing through points $A$ and $K$ intersects sides $AB$ and $AC$ at points $M$ and $N$, respectively. The line passing through point $A$ and parallel to line $BC$ intersects the circumcircles of triangles $AHM$ and $AHN$ for second time, respectively, at points $X$ and $Y$. Prove that $ | X Y | = | BC |$.
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.
[i]Proposed by Holden Mui[/i]
Let $ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$. Let the height from $A$ cut its side
$BC$ at $D$. Let $I, I_B, I_C$ be the incenters of triangles $ABC, ABD, ACD$ respectively. Let also
$EB, EC$ be the excenters of $ABC$ with respect to vertices $B$ and $C$ respectively. If $K$ is the
point of intersection of the circumcircles of $E_CIB_I$ and $E_BIC_I$, show that $KI$ passes through
the midpoint $M$ of side $BC$.
Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$.
[i]Proposed by David Monk, United Kingdom[/i]
In the non-isosceles triangle $ABC$,$D$ is the midpoint of side $BC$,$E$ is the midpoint of side $CA$,$F$ is the midpoint of side $AB$.The line(different from line $BC$) that is tangent to the inscribed circle of triangle $ABC$ and passing through point $D$ intersect line $EF$ at $X$.Define $Y,Z$ similarly.Prove that $X,Y,Z$ are collinear.
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.
Let $ABC$ be a triangle whose angle at $A$ is right, and let $D$ be the foot of the altitude from $A$. A variable point $M$ traces the interior of the minor arc $AB$ of the circle $ABC$. The internal bisector of the angle $DAM$ crosses $CM$ at $N$. The line through $N$ and perpendicular to $CM$ crosses the line $AD$ at $P$. Determine the locus of the point where the line $BN$ crosses the line $CP$.
[i]* * *[/i]
Given an $\triangle ABC$ isoceles with base $BC$ we note with $M$ the midpoint of said base. Let $X$ be any point on the shortest arc $AM$ of the circumcircle of $\triangle ABM$ and let $T$ be a point on the inside $\angle BMA$ such that $\angle TMX = 90^o$ and $TX = BX$. Show that $\angle MTB - \angle CTM$ does not depend on $X$.
Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent.
Maxim Didin
In a non isoceles triangle $ABC$, let the perpendicular bisector of $[BC]$ intersect $(ABC)$ at $M$ and $N$ respectively. Let the midpoints of $[AM]$ and $[AN]$ be $K$ and $L$ respectively. Let $(ABK)$ and $(ABL)$ intersect $AC$ again at $D$ and $E$ respectively, let $(ACK)$ and $(ACL)$ intersect $AB$ again at $F$ and $G$ respectively.
Prove that the lines $DF$, $EG$ and $MN$ are concurrent.
Let $ABCD$ be a square, and let $P$ be a point on the minor arc $CD$ of its circumcircle. The lines $PA, PB$ meet the diagonals $BD, AC$ at points $K, L$ respectively. The points $M, N$ are the projections of $K, L$ respectively to $CD$, and $Q$ is the common point of lines $KN$ and $ML$. Prove that $PQ$ bisects the segment $AB$.
Consider an acute triangle $ABC$ with $AB>AC$ inscribed in a circle of center $O$. From the midpoint $D$ of side $BC$ we draw line $(\ell)$ perpendicular to side $AB$ that intersects it at point $E$. If line $AO$ intersects line $(\ell)$ at point $Z$, prove that points $A,Z,D,C$ are concyclic.
Given an acute triangle $ABC$ and $(O)$ be its circumcircle. Let $G$ be the point on arc $BC$ that doesn't contain $O$ of the circumcircle $(I)$ of triangle $OBC$. The circumcircle of $ABG$ intersects $AC$ at $E$ and circumcircle of $ACG$ intersects $AB$ at $F$ ($E\ne A, F\ne A$).
a) Let $K$ be the intersection of $BE$ and $CF$. Prove that $AK,BC,OG$ are concurrent.
b) Let $D$ be a point on arc $BOC$ (arc $BC$ containing $O$) of $(I)$. $GB$ meets $CD$ at $M$ , $GC$ meets $BD$ at $N$. Assume that $MN$ intersects $(O)$ at $P$ nad $Q$. Prove that when $G$ moves on the arc $BC$ that doesn't contain $O$ of $(I)$, the circumcircle $(GPQ)$ always passes through two fixed points.
Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that
\[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3,
\]
denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel.
[i]Nikolai Ivanov Beluhov, Bulgaria[/i]
Let $ABC$ be a triangle. Let $M$ and $N$ be the points in which the median and the angle bisector, respectively, at $A$ meet the side $BC$. Let $Q$ and $P$ be the points in which the perpendicular at $N$ to $NA$ meets $MA$ and $BA$, respectively. And $O$ the point in which the perpendicular at $P$ to $BA$ meets $AN$ produced.
Prove that $QO$ is perpendicular to $BC$.
In triangle $\triangle ABC$ , $M, N$ are midpoints of $AC,AB$ respectively. Assume that $BM,CN$ cuts $(ABC)$ at $M',N'$ respectively. Let $X$ be on the extention of $BC$ from $B$ st $\angle N'XB=\angle ACN$. And define $Y$ similarly on the extention of $BC$ from $C$. Prove that $AX=AY$.
Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.
Let $ABCD$ be a cyclic quadrilateral, with circumcircle $\omega$ and circumcenter $O$. Let $AB$ intersect $CD$ at $E$, $AD$ intersect $BC$ at $F$, and $AC$ intersect $BD$ at $G$.
The points $A_1, B_1, C_1, D_1$ are chosen on rays $GA$, $GB$, $GC$, $GD$ such that:
$\bullet$ $\displaystyle \frac{GA_1}{GA} = \frac{GB_1}{GB} = \frac{GC_1}{GC} = \frac{GD_1}{GD}$
$\bullet$ The points $A_1, B_1, C_1, D_1, O$ lie on a circle.
Let $A_1B_1$ intersect $C_1D_1$ at $K$, and $A_1D_1$ intersect $B_1C_1$ at $L$. Prove that the image of the circle $(A_1B_1C_1D_1)$ under inversion about $\omega$ is a line passing through the midpoints of $KE$ and $LF$.
[i]Proposed by Anzo Teh Zhao Yang & Ivan Chan Kai Chin[/i]
$ABCD$ is a cyclic quadrilateral with $AC \perp BD$; $AC$ meets $BD$ at $E$. Prove that \[ EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2 \]
where $R$ is the radius of the circumscribing circle.
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$. Let $F_A, F_B, F_C$ and $F_D$ be the midpoints of arcs $AB, BC, CD$ and $DA$ of $\omega$. Let $I_A, I_B, I_C$ and $I_D$ be the incenters of triangles $DAB, ABC, BCD$ and $CDA$, respectively.
Let $\omega_A$ denote the circle that is tangent to $\omega$ at $F_A$ and also tangent to line segment $CD$. Similarly, let $\omega_C$ denote the circle that is tangent to $\omega$ at $F_C$ and tangent to line segment $AB$.
Finally, let $T_B$ denote the second intersection of $\omega$ and circle $F_BI_BI_C$ different from $F_B$, and let $T_D$ denote the second intersection of $\omega$ and circle $F_DI_DI_A$.
Prove that the radical axis of circles $\omega_A$ and $\omega_C$ passes through points $T_B$ and $T_D$.
Let $\triangle ABC$ be an acute triangle, where $H$ is the orthocenter and $D,E,F$ are the feet of the altitudes from $A,B,C$ respectively. A circle tangent to $(DEF)$ at $D$ intersects the line $EF$ at $P$ and $Q$. Let $R$ and $S$ be the second intersection points of the circumcircle of triangle $\triangle BHC$ with $PH$ and $QH$, respectively. Let $T$ be the point on the line $BC$ such that $AT\perp EF$.
Prove that the points $R,S,D,T$ are concyclic.
Prove that the product of the sides of a quadrilateral inscribed in a circle with radius $1$ does not exceed $4$.