Found problems: 533
Let convex quadrilateral $ ABCD$ be inscribed in a circle centers at $ O.$ The opposite sides $ BA,CD$ meet at $ H$, the diagonals $ AC,BD$ meet at $ G.$ Let $ O_{1},O_{2}$ be the circumcenters of triangles $ AGD,BGC.$ $ O_{1}O_{2}$ intersects $ OG$ at $ N.$ The line $ HG$ cuts the circumcircles of triangles $ AGD,BGC$ at $ P,Q$, respectively. Denote by $ M$ the midpoint of $ PQ.$ Prove that $ NO \equal{} NM.$
A circle $k$ passes through the vertices $B, C$ of a scalene triangle $ABC$. $k$ meets the extensions of $AB, AC$ beyond $B, C$ at $P, Q$ respectively. Let $A_1$ is the foot the altitude drop from $A$ to $BC$. Suppose $A_1P=A_1Q$. Prove that $\widehat{PA_1Q}=2\widehat{BAC}$.
Let $ABC$ be an acute-angled triangle with $AC>AB$, and $\Omega$ is your circumcircle. Let $P$ be the midpoint of the arc $BC$ of $\Omega$ (not containing $A$) and $Q$ be the midpoint of the arc $BC$ of $\Omega$(containing the point $A$). Let $M$ be the foot of perpendicular of $Q$ on the line $AC$. Prove that the circumcircle of $\triangle AMB$ cut the segment $AP$ in your midpoint.
In convex hexagon $ABCDEF$, $\angle A \cong \angle B$, $\angle C \cong \angle D$, and $\angle E \cong \angle F$. Prove that the perpendicular bisectors of $\overline{AB}$, $\overline{CD}$, and $\overline{EF}$ pass through a common point.
[i]Proposed by Lewis Chen[/i]
Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. Prove that if $BE = BR$, then $\angle ELK = \tfrac{1}{2} \angle BCD$.
[i]Proposed by Evan Chen[/i]
Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$.
[i]Proposed by Carlos Yuzo Shine, Brazil[/i]
Let $ABCD$ be a convex quadrilateral and let $M$ be the midpoint of side $AB$. The circle passing through $D$ and tangent to $AB$ at $A$ intersects the segment $DM$ at $E$. The circle passing through $C$ and tangent to $AB$ at $B$ intersects the segment $CM$ at $F$. Suppose that the lines $AF$ and $BE$ intersect at a point which belongs to the perpendicular bisector of side $AB$. Prove that $A$, $E$, and $C$ are collinear if and only if $B$, $F$, and $D$ are collinear.
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 $BM = CM$.
Let $ABC$ be an acute non isosceles triangle. The angle bisector of angle $A$ meets again the circumcircle of the triangle $ABC$ in $D$. Let $O$ be the circumcenter of the triangle $ABC$. The angle bisectors of $\angle AOB$, and $\angle AOC$ meet the circle $\gamma$ of diameter $AD$ in $P$ and $Q$ respectively. The line $PQ$ meets the perpendicular bisector of $AD$ in $R$. Prove that $AR // BC$.
$ABC$ is acute-angled. $D$ s a variable point on the side BC. $O_1$ is the circumcenter of $ABD$, $O_2$ is the circumcenter of $ACD$, and $O$ is the circumcenter of $AO_1O_2$. Find the locus of $O$.
Let $\triangle ABC$ which $\angle ABC$ are right angle, Let $D$ be point on $AB$ ( $D \neq A , B$ ), Let $E$ be point on line $AB$ which $B$ is the midpoint of $DE$, Let $I$ be incenter of $\triangle ACE$ and $J$ be $A$-excenter of $\triangle ACD$. Prove that perpendicular bisector of $BC$ bisects $IJ$
In acute triangle $ABC$, an arbitrary point $P$ is chosen on altitude $AH$. Points $E$ and $F$ are the midpoints of sides $CA$ and $AB$ respectively. The perpendiculars from $E$ to $CP$ and from $F$ to $BP$ meet at point $K$. Prove that $KB = KC$.
Let $ABC$ be an acute-angled triangle. The line through $C$ perpendicular to $AC$ meets the external angle bisector of $\angle ABC$ at $D$. Let $H$ be the foot of the perpendicular from $D$ onto $BC$. The point $K$ is chosen on $AB$ so that $KH \parallel AC$. Let $M$ be the midpoint of $AK$. Prove that $MC = MB + BH$.
Giorgi Arabidze, Georgia,
A line segment of length 1 is given on the plane. Show that a line segment of length $\sqrt{2010}$ can be constructed using only a straightedge and a compass.
The vertices of a square sheet of paper are $ A$, $ B$, $ C$, $ D$. The sheet is folded in a way that the point $ D$ is mapped to the point $ D'$ on the side $ BC$. Let $ A'$ be the image of $ A$ after the folding, and let $ E$ be the intersection point of $ AB$ and $ A'D'$. Let $ r$ be the inradius of the triangle $ EBD'$. Prove that $ r\equal{}A'E$.
For a rectangle $ABCD$ which is not a square, there is $O$ such that $O$ is on the perpendicular bisector of $BD$ and $O$ is in the interior of $\triangle BCD$. Denote by $E$ and $F$ the second intersections of the circle centered at $O$ passing through $B, D$ and $AB, AD$. $BF$ and $DE$ meets at $G$, and $X, Y, Z$ are the foots of the perpendiculars from $G$ to $AB, BD, DA$. $L, M, N$ are the foots of the perpendiculars from $O$ to $CD, BD, BC$. $XY$ and $ML$ meets at $P$, $YZ$ and $MN$ meets at $Q$. Prove that $BP$ and $DQ$ are parallel.
Given a parallelogram $ABCD$ with $AB<BC$, show that the circumcircles of the triangles $APQ$ share a second common point (apart from $A$) as $P,Q$ move on the sides $BC,CD$ respectively s.t. $CP=CQ$.
Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$, and let $m_i$ be the multiplicity of $d_i$, i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$. Prove that $\sum_{i=1}^k m_i^2\leq n^3-n^2$.
Let $AL$ and $BK$ be the angle bisectors in a non-isosceles triangle $ABC,$ where $L$ lies on $BC$ and $K$ lies on $AC.$ The perpendicular bisector of $BK$ intersects the line $AL$ at $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK.$ Prove that $LN=NA.$
Let $ABC$ be an acute-angled triangle with $H$ orthocenter, $O$ circumcenter. $[AH]$'s perpendicular bisector intersects with $[AB]$ and $[AC]$ at $D$ and $E$ respectively. Prove that
$$\angle ADE =\angle BDO$$
Triangle ABC has incircle w centered as S that touches the sides BC,CA and AB at P,Q and R respectively. AB isn't equal AC, the lines QR and BC intersects at point M, the circle that passes through points B and C touches the circle w at point N, circumcircle of triangle MNP intersects with line AP at L (L isn't equal to P). Then prove that S,L and M lie on the same line
Let $M, N$ be the midpoints of sides $AB, AC$ respectively of a triangle $ABC$. The perpendicular bisector to the bisectrix $AL$ meets the bisectrixes of angles $B$ and $C$ at points $P$ and $Q$ respectively. Prove that the common point of lines $PM$ and $QN$ lies on the tangent to the circumcircle of $ABC$ at $A$.
Triangle $ABC$ is inscribed in circle $O$. Tangent $PD$ is drawn from $A$, $D$ is on ray $BC$, $P$ is on ray $DA$. Line $PU$ ($U \in BD$) intersects circle $O$ at $Q$, $T$, and intersect $AB$ and $AC$ at $R$ and $S$ respectively. Prove that if $QR=ST$, then $PQ=UT$.
A convex quadrilateral $ABCD$ has an incircle. In each corner a circle is inscribed that also externally touches the two circles inscribed in the adjacent corners. Show that at least two circles have the same size.
In the quadrilateral $ABCD$, $AB = BC$ . The point $E$ lies on the line $AB$ is such that $BD= BE$ and $AD \perp DE$. Prove that the perpendicular bisectors to segments $AD, CD$ and $CE$ intersect at one point.