Found problems: 3585
A cyclic quadrilateral $ABCD$ has circumcircle $\Gamma$, and $AB+BC=AD+DC$. Let $E$ be the midpoint of arc $BCD$, and $F (\neq C)$ be the antipode of $A$ [i]wrt[/i] $\Gamma$. Let $I,J,K$ be the incenter of $\triangle ABC$, the $A$-excenter of $\triangle ABC$, the incenter of $\triangle BCD$, respectively.
Suppose that a point $P$ satisfies $\triangle BIC \stackrel{+}{\sim} \triangle KPJ$. Prove that $EK$ and $PF$ intersect on $\Gamma.$
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
Let $ABC$ be an acute scalene triangle, let $D$ be a point on the $A$-altitude, and let the circle with diameter $AD$ meet $AC$, $AB$, and the circumcircle of $ABC$ at $E$, $F$, $G$, respectively. Let $O$ be the circumcenter of $ABC$, let $AO$ meet $EF$ at $T$, and suppose the circumcircles of $ABC$ and $GTO$ meet at $X \neq G$. Then, prove that $AX$, $DG$, and $EF$ concur.
Let $ABCD$ be a convex quadrilateral with $\angle DAB=\angle BCD=90^{\circ}$ and $\angle ABC> \angle CDA$. Let $Q$ and $R$ be points on segments $BC$ and $CD$, respectively, such that line $QR$ intersects lines $AB$ and $AD$ at points $P$ and $S$, respectively. It is given that $PQ=RS$.Let the midpoint of $BD$ be $M$ and the midpoint of $QR$ be $N$.Prove that the points $M,N,A$ and $C$ lie on a circle.
A quadrilateral $ABCD$ without parallel sides is inscribed in a circle $\omega$. We draw a line $\ell_a \parallel BC$ through the point $A$, a line $\ell_b \parallel CD$ through the point $B$, a line $\ell_c \parallel DA$ through the point $C$, and a line $\ell_d \parallel AB$ through the point $D$. Suppose that the quadrilateral whose successive sides lie on these four straight lines is inscribed in a circle $\gamma$ and that $\omega$ and $\gamma$ intersect in points $E$ and $F$. Show that the lines $AC, BD$ and $EF$ intersect in one point.
[i]Proposed by A. Kuznetsov[/i]
The altitudes of the acute-angled non-isosceles triangle $ ABC $ intersect at the point $ H $. On the segment $ C_1H $, where $ CC_1 $ is the altitude of the triangle, the point $ K $ is marked. Points $ L $ and $ M $ are the feet of perpendiculars from point $ K $ on straight lines $ AC $ and $ BC $, respectively. The lines $ AM $ and $ BL $ intersect at $ N $. Prove that $ \angle ANK = \angle HNL $.
Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$. A straight line $L$ is parallel to $BC$ and tangent to the incircle. Suppose $L$ intersects $IO$ at $X$, and select $Y$ on $L$ such that $YI$ is perpendicular to $IO$. Prove that $A$, $X$, $O$, $Y$ are cyclic.
[i]Proposed by Telv Cohl[/i]
Triangle $ABC$ is given with its centroid $G$ and cicumcentre $O$ is such that $GO$ is perpendicular to $AG$. Let $A'$ be the second intersection of $AG$ with circumcircle of triangle $ABC$. Let $D$ be the intersection of lines $CA'$ and $AB$ and $E$ the intersection of lines $BA'$ and $AC$. Prove that the circumcentre of triangle $ADE$ is on the circumcircle of triangle $ABC$.
Let $I$ and $H$ be the incentre and orthocentre of triangle $ABC$ respectively. Let $P,Q$ be the midpoints of $AB,AC$. The rays $PI,QI$ intersect $AC,AB$ at $R,S$ respectively. Suppose that $T$ is the circumcentre of triangle $BHC$. Let $RS$ intersect $BC$ at $K$. Prove that $A,I$ and $T$ are collinear if and only if $[BKS]=[CKR]$.
[i]Shen Wunxuan[/i]
$ABCD$ is a cyclic quadrilateral in which $AB=4$, $BC=3$, $CD=2$, and $AD=5$. Diagonals $AC$ and $BD$ intersect at $X$. A circle $\omega$ passes through $A$ and is tangent to $BD$ at $X$. $\omega$ intersects $AB$ and $AD$ at $Y$ and $Z$ respectively. Compute $YZ/BD$.
Given is a triangle $ABC$ with circumcentre $O$. The circumcircle of triangle $AOC$ intersects side $BC$ at $D$ and side $AB$ at $E$. Prove that the triangles $BDE$ and $AOC$ have circumradiuses of equal length.
Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.
[i]Robin Son[/i]
Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively.
a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$.
b) Does the converse implication also always hold?
Let $ABC$ be a triangle. Let $T$ be the point of tangency of the circumcircle of triangle $ABC$ and the $A$-mixtilinear incircle. The incircle of triangle $ABC$ has center $I$ and touches sides $BC,CA$ and $AB$ at points $D,E$ and $F,$ respectively. Let $N$ be the midpoint of line segment $DF.$ Prove that the circumcircle of triangle $BTN,$ line $TI$ and the perpendicular from $D$ to $EF$ are concurrent.
[i]Proposed by Diaconescu Tashi, Romania[/i]
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 triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$, and that $\angle AXO = \angle OXA'$.
[i]Proposed by Sammy Luo[/i]
Let $ ABCD$ be a cyclic quadrilateral and let $ E$ and $ F$ be the feet of perpendiculars from the intersection of diagonals $ AC$ and $ BD$ to $ AB$ and $ CD$, respectively. Prove that $ EF$ is perpendicular to the line through the midpoints of $ AD$ and $ BC$.
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 $\angle ATB$.
In triangle $ABC$, $AB = AC$. Point $D$ is the midpoint of side $BC$. Point $E$ lies outside the triangle $ABC$ such that $CE \perp AB$ and $BE = BD$. Let $M$ be the midpoint of segment $BE$. Point $F$ lies on the minor arc $\widehat{AD}$ of the circumcircle of triangle $ABD$ such that $MF \perp BE$. Prove that $ED \perp FD.$
[asy]
defaultpen(fontsize(10)); size(6cm);
pair A = (3,10), B = (0,0), C = (6,0), D = (3,0), E = intersectionpoints( Circle(B, 3), C--(C+100*dir(B--A)*dir(90)) )[1], M = midpoint(B--E), F = intersectionpoints(M--(M+50*dir(E--B)*dir(90)), circumcircle(A,B,D))[0];
dot(A^^B^^C^^D^^E^^M^^F);
draw(B--C--A--B--E--D--F--M^^circumcircle(A,B,D));
pair point = extension(M,F,A,D);
pair[] p={A,B,C,D,E,F,M};
string s = "A,B,C,D,E,F,M";
int size = p.length;
real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;}
d[4] = -50;
string[] k= split(s,",");
for(int i = 0;i<p.length;++i) {
label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i]));
}[/asy]
Circles $O_1$ and $O_2$ interest at two points $ B$ and $ C,$ and $ BC$ is the diameter of circle $O_1.$ Construct a tangent line of circle $O_1$ at $ C$ and intersecting circle $O_2$ at another point $ A.$ We join $ AB$ to intersect circle $O_1$ at point $ E,$ then join $ CE$ and extend it to intersect circle $O_2$ at point $ F.$ Assume $ H$ is an arbitrary point on line segment $ AF.$ We join $ HE$ and extend it to intersect circle $O_1$ at point $ G,$ and then join $ BG$ and extend it to intersect the extend line of $ AC$ at point $ D.$ Prove that \[ \frac{AH}{HF} = \frac{AC}{CD}.\]
Let $ AB$ be the diameter of semicircle $O$ ,
$C, D $ be points on the arc $AB$,
$P, Q$ be respectively the circumcenter of $\triangle OAC $ and $\triangle OBD $ .
Prove that:$CP\cdot CQ=DP \cdot DQ$.[asy]
import cse5; import olympiad; unitsize(3.5cm); dotfactor=4; pathpen=black;
real h=sqrt(55/64);
pair A=(-1,0), O=origin, B=(1,0),C=shift(-3/8,h)*O,D=shift(4/5,3/5)*O,P=circumcenter(O,A,C), Q=circumcenter(O,D,B);
D(arc(O,1,0,180),darkgreen);
D(MP("A",A,W)--MP("C",C,N)--MP("P",P,SE)--MP("D",D,E)--MP("Q",Q,E)--C--MP("O",O,S)--D--MP("B",B,E)--cycle,deepblue);
D(O);
[/asy]
Diagonals of a quadrilateral $ABCD$ are equal and meet at point $O$. The perpendicular bisectors to segments $AB$ and $CD$ meet at point $P$, and the perpendicular bisectors to $BC$ and $AD$ meet at point $Q$. Find angle $\angle POQ$.
by A.Zaslavsky
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 $\angle BAC=90^{\circ}$.
Let $ABC$ be an acute triangle and $M$ be the midpoint of $AB$. A circle through the points $B$ and $C$ intersects the segments $CM$ and $BM$ at points $P$ and $Q$ respectively. Point $K$ is symmetric to $P$ with respect to point $M$. The circumcircles of $\triangle AKM$ and $\triangle CQM$ intersect for the second time at $X$. The circumcircles of $\triangle AMC$ and $\triangle KMQ$ intersect for the second time at $Y$. The segments $BP$ and $CQ$ intersect at point $T$. Prove that the line $MT$ is tangent to the circumcircle of $\triangle MXY$.
We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point).
(a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$.
(b) Find the largest possible size of a very set not contained in any line.
(Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.)
[i]Proposed by Sammy Luo[/i]