Found problems: 3585
$\triangle ABC$ is an equilateral triangle. $D$ is a point on $\overline{AC}$, and $E$ is a point on $\overline{BD}$. Let $P$ and $Q$ be the circumcenters of $\triangle ABD$ and $\triangle AED$, respectively. Prove that $ \triangle EPQ$ is an equilateral triangle if and only if $ \overline{AB} \perp \overline{CE}$.
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 $\triangle ABC$, the incircle $\omega$ has center $I$ and is tangent to $\overline{CA}$ and $\overline{AB}$ at $E$ and $F$ respectively. The circumcircle of $\triangle{BIC}$ meets $\omega$ at $P$ and $Q$. Lines $AI$ and $BC$ meet at $D$, and the circumcircle of $\triangle PDQ$ meets $\overline{BC}$ again at $X$. Suppose that $EF = PQ = 16$ and $PX + QX = 17$. Then $BC^2$ can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$.
[i]Proposed by Ankan Bhattacharya and Michael Ren
Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other.
$\emph{Slovakia}$
$AB < AC$ on $\triangle ABC$. The midpoint of arc $BC$ which doesn't include $A$ is $T$ and which includes $A$ is $S$. On segment $AB,AC$, $D,E$ exist so that $DE$ and $BC$ are parallel. The outer angle bisector of $\angle ABE$ and $\angle ACD$ meets $AS$ at $P$ and $Q$. Prove that the circumcircle of $\triangle PBE$ and $\triangle QCD$ meets on $AT$.
Quadrilateral $ABCD$ with perpendicular diagonals is inscribed in a circle with centre $O$. The tangents to this circle at $A$ and $C$ together with line $BD$ form the triangle $\Delta$. Prove that the circumcircles of $BOD$ and $\Delta$ are tangent.
[hide=Additional information for Junior League]Show that this point lies belongs to $\omega$, the circumcircle of $OAC$[/hide]
[i]Proposed by A. Kuznetsov[/i]
Let $ABC$ be a triangle ($\angle A\neq 90^\circ$). $BE,CF$ are the altitudes of the triangle. The bisector of $\angle A$ intersects $EF,BC$ at $M,N$. Let $P$ be a point such that $MP\perp EF$ and $NP\perp BC$. Prove that $AP$ passes through the midpoint of $BC$.
[i]Proposed by Iman Maghsoudi, Hooman Fattahi[/i]
Let $\omega$ be the circumcircle of acute triangle $ABC$. Two tangents of $\omega$ from $B$ and $C$ intersect at $P$, $AP$ and $BC$ intersect at $D$. Point $E$, $F$ are on $AC$ and $AB$ such that $DE \parallel BA$ and $DF \parallel CA$.
(1) Prove that $F,B,C,E$ are concyclic.
(2) Denote $A_{1}$ the centre of the circle passing through $F,B,C,E$. $B_{1}$, $C_{1}$ are difined similarly. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent.
Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.
Let $ABCD$ be a convex cyclic quadrilateral with diagonals $AC$ and $BD$. Each of the four vertixes are reflected across the diagonal on which the do not lie.
(a) Investigate when the four points thus obtained lie on a straight line and give as simple an equivalent condition as possible to the cyclic quadrilateral $ABCD$ for it.
(b) Show that in all other cases the four points thus obtained lie on one circle.
(Theresia Eisenkölbl)
Let $ABCD$ be a convex quadrileteral such that $m(\widehat{ABD})=40^\circ$, $m(\widehat{DBC})=70^\circ$, $m(\widehat{BDA})=80^\circ$, and $m(\widehat{BDC})=50^\circ$. What is $m(\widehat{CAD})$?
$
\textbf{(A)}\ 25^\circ
\qquad\textbf{(B)}\ 30^\circ
\qquad\textbf{(C)}\ 35^\circ
\qquad\textbf{(D)}\ 38^\circ
\qquad\textbf{(E)}\ 40^\circ
$
Let ${ABC}$ be a triangle. Line [i]l[/i] is parallel to ${BC}$ and it respectively intersects side ${AB}$ at point ${D}$, side ${AC}$ at point ${E}$, and the circumcircle of the triangle ${ABC}$ at points ${F}$ and ${G}$, where points ${F,D,E,G}$ lie in this order on [i]l[/i]. The circumcircles of triangles ${FEB}$ and ${DGC}$ intersect at points ${P}$ and ${Q}$. Prove that points ${A, P,Q}$ are collinear.
(Slovakia)
Consider a cyclic quadrilateral $ABCD$, and let $S$ be the intersection of $AC$ and $BD$.
Let $E$ and $F$ the orthogonal projections of $S$ on $AB$ and $CD$ respectively.
Prove that the perpendicular bisector of segment $EF$ meets the segments $AD$ and $BC$ at their midpoints.
Let $y = x^2 + ax + b$ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$.
Points$ D, E, F$ are chosen on the sides $AB$, $BC$, $AC$ of a triangle $ABC$, so that $DE = BE$ and $FE = CE$. Prove that the centre of the circle circumscribed around triangle $ADF$ lies on the bisectrix of angle $DEF$.
In cyclic quadrilateral $ABCD$, $AB>BC$, $AD>DC$, $I,J$ are the incenters of $\triangle ABC$,$\triangle ADC$ respectively. The circle with diameter $AC$ meets segment $IB$ at $X$, and the extension of $JD$ at $Y$. Prove that if the four points $B,I,J,D$ are concyclic, then $X,Y$ are the reflections of each other across $AC$.
Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.
Consider a cyclic quadrilateral $ABCD$. The extensions of its sides $AB,DC$ meet at the point $P$ and the extensions of its sides $AD,BC$ meet at the point $Q$. Suppose $\quad QE,QF$ are tangents to the circumcircle of quadrilateral $ABCD$ at $E,F$ respectively. Show that $P,E,F$ are collinear.
Let $ABC$ be a triangle with altitudes $AD, BE, CF$ and orthocenter $H$. The perpendicular bisector of $HD$ meets $EF$ at $P$ and $N$ is the center of the nine-point circle. Let $L$ be a point on the circumcircle of $ABC$ such that $\angle PLN=90^{\circ}$ and $A, L$ are in distinct sides of the line $PN$. Show that $ANDL$ is cyclic.
Let $\omega$ be the circumcircle of $ABC$, and $KL$ be the diameter of $\omega$ passing through $M$ midpoint of $AB$ ($K,C$ lies on different sides of $AB$). A circle passing through $L$ and $M$ meets $CK$ at points $P$ and $Q$ ($Q$ lies on $KP$). Let $LQ$ meet the circumcircle of $KMQ$ again at $R$. Prove that $APBR$ is cyclic.
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
Let $ABC$ be a triangle with circumcircle $\omega$ and circumcenter $O$. The tangent line to from $A$ to $\omega$ intersects $BC$ at $K$. The tangent line to from $B$ to $\omega$ intersects $AC$ at $L$. Let $M,N$ be the midpoints of $AK,BL$ respectively. The line $MN$ is named by $\alpha$. The feet of perpendicular from $A,B,C$ to the edges of $\triangle ABC$ are named by $D,E,F$ respectively. The perpendicular bisectors of $EF,DF,DE$ intersect $\alpha$ at $X,Y,Z$ respectively. Let $AD,BE,CF$ intersect $\omega$ again at $D',E',F'$ respectively. If $H$ is the orthocenter of $ABC$, prove that the lines $XD',YE',ZF',OH$ are concurrent.
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
Let $ABC$ be a triangle. Points $D, E, F$ are on segments $BC$, $CA$, $AB$, respectively. Suppose that $AF = 10$, $F B = 10$, $BD = 12$, $DC = 17$, $CE = 11$, and $EA = 10$. Suppose that the circumcircles of $\vartriangle BFD$ and $\vartriangle CED$ intersect again at $X$. Find the circumradius of $\vartriangle EXF$.
We are given an acute triangle $ABC$ with $AB\neq AC$. Let $D$ be a point of $BC$ such that $DA$ is tangent to the circumcircle of $ABC$. Let $E$ and $F$ be the circumcenters of triangles $ABD$ and $ACD$, respectively, and let $M$ be the midpoints $EF$. Prove that the line tangent to the circumcircle of $AMD$ through $D$ is also tangent to the circumcircle of $ABC$.
[i]Proposed by Patrik Bak, Slovakia[/i]