Found problems: 533
Given is a triangle $ABC$ with barycenter $G$ and circumcenter $O$.The perpendicular bisectors of $GA,GB,GC$ intersect at $A_1,B_1,C_1$.Show that $O$ is the barycenter of $\triangle{A_1B_1C_1}$.
Let $ ABC$ be a tringle and let $ P$ be an interior point such that $ \angle BPC \equal{} 90 ,\angle BAP \equal{} \angle BCP$.Let $ M,N$ be the mid points of $ AC,BC$ respectively.Suppose $ BP \equal{} 2PM$.Prove that $ A,P,N$ are collinear.
In convex quadrilateral $ ABCD$, $ AB \equal{} BC$ and $ AD \equal{} DC$. Point $ E$ lies on segment $ AB$ and point $ F$ lies on segment $ AD$ such that $ B$, $ E$, $ F$, $ D$ lie on a circle. Point $ P$ is such that triangles $ DPE$ and $ ADC$ are similar and the corresponding vertices are in the same orientation (clockwise or counterclockwise). Point $ Q$ is such that triangles $ BQF$ and $ ABC$ are similar and the corresponding vertices are in the same orientation. Prove that points $ A$, $ P$, $ Q$ are collinear.
Let $ABCD$ be a rectangle. Determine the set of all points $P$ from the region between the parallel lines $AB$ and $CD$ such that $\angle APB=\angle CPD$.
The perpendicular bisector of the bisector $BL$ of the triangle $ABC$ intersects the bisectors of its external angles $A$ and $C$ at points $P$ and $Q$, respectively. Prove that the circle circumscribed around the triangle $PBQ$ is tangent to the circle circumscribed around the triangle $ABC$.
Let $ABC$ be an acute-angled, not equilateral triangle, where vertex $A$ lies on the perpendicular bisector of the segment $HO$, joining the orthocentre $H$ to the circumcentre $O$. Determine all possible values for the measure of angle $A$.
(U.S.A. - 1989 IMO Shortlist)
Consider a circle $S$, and a point $P$ outside it. The tangent lines from $P$ meet $S$ at $A$ and $B$, respectively. Let $M$ be the midpoint of $AB$. The perpendicular bisector of $AM$ meets $S$ in a point $C$ lying inside the triangle $ABP$. $AC$ intersects $PM$ at $G$, and $PM$ meets $S$ in a point $D$ lying outside the triangle $ABP$. If $BD$ is parallel to $AC$, show that $G$ is the centroid of the triangle $ABP$.
[i]Arnoldo Aguilar (El Salvador)[/i]
Let be a point $ P $ in the interior of a triangle $ ABC $ such that $ BP=AC, M $ be the middlepoint of the segment $ AP,
R $ be the middlepoint of $ BC $ and $ E $ be the intersection of $ BP $ with $ AC. $ Prove that the bisector of $ \angle BEA $ is perpendicular on $ MR $
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]
Given a triangle $ABC$ inscribed by circumcircle $(O)$. The angles at $B,C$ are acute angle. Let $M$ on the arc $BC$ that doesn't contain $A$ such that $AM$ is not perpendicular to $BC$. $AM$ meets the perpendicular bisector of $BC$ at $T$. The circumcircle $(AOT)$ meets $(O)$ at $N$ ($N\ne A$).
a) Prove that $\angle{BAM}=\angle{CAN}$.
b) Let $I$ be the incenter and $G$ be the foor of the angle bisector of $\angle{BAC}$. $AI,MI,NI$ intersect $(O)$ at $D,E,F$ respectively. Let ${P}=DF\cap AM, {Q}=DE\cap AN$. The circle passes through $P$ and touches $AD$ at $I$ meets $DF$ at $H$ ($H\ne D$).The circle passes through $Q$ and touches $AD$ at $I$ meets $DE$ at $K$ ($K\ne D$). Prove that the circumcircle $(GHK)$ touches $BC$.
Given a triangle $ABC$, in which $\angle B> 90^o$. Perpendicular bisector of the side $AB$ intersects the side $AC$ at the point $M$, and the perpendicular bisector of the side $AC$ intersects the extension of the side $AB$ beyond the vertex $B$ at point $N$. It is known that the segments $MN$ and $BC$ are equal and intersect at right angles. Find the values of all angles of triangle $ABC$.
As shown in the figure, quadrilateral $ ABCD$ is inscribed in a circle with $ AC$ as its diameter, $ BD \perp AC,$ and $ E$ the intersection of $ AC$ and $ BD.$ Extend line segment $ DA$ and $ BA$ through $ A$ to $ F$ and $ G$ respectively, such that $ DG \parallel{} BF.$ Extend $ GF$ to $ H$ such that $ CH \perp GH.$ Prove that points $ B, E, F$ and $ H$ lie on one circle.
[asy]
defaultpen(linewidth(0.8)+fontsize(10));size(150);
real a=4, b=6.5, c=9, d=a*c/b, g=14, f=sqrt(a^2+b^2)*sqrt(a^2+d^2)/g;
pair E=origin, A=(0,a), B=(-b,0), C=(0,-c), D=(d,0), G=A+g*dir(B--A), F=A+f*dir(D--A), M=midpoint(G--C);
path c1=circumcircle(A,B,C), c2=Circle(M, abs(M-G));
pair Hf=F+10*dir(G--F), H=intersectionpoint(F--Hf, c2);
dot(A^^B^^C^^D^^E^^F^^G^^H);
draw(c1^^c2^^G--D--C--A--G--F--D--B--A^^F--H--C--B--F);
draw(H--B^^F--E^^G--C, linetype("2 2"));
pair point= E;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$F$", F, dir(point--F));
label("$G$", G, dir(point--G));
label("$H$", H, dir(point--H));
label("$E$", E, NE);[/asy]
Consider a rhombus $ABCD$ with center $O$. A point $P$ is given inside the rhombus, but not situated on the diagonals. Let $M,N,Q,R$ be the projections of $P$ on the sides $(AB), (BC), (CD), (DA)$, respectively. The perpendicular bisectors of the segments $MN$ and $RQ$ meet at $S$ and the perpendicular bisectors of the segments $NQ$ and $MR$ meet at $T$. Prove that $P, S, T$ and $O$ are the vertices of a rectangle.
Starting from a given cyclic quadrilateral $\mathcal{Q}_0$, a sequence of quadrilaterals is constructed so that $\mathcal{Q}_{k + 1}$ is the circumscribed quadrilateral of $\mathcal{Q}_k$ for $k = 0,1,\dots$. The sequence terminates when a quadrilateral is reached that is not cyclic. (The circumscribed quadrilateral of a cylic quadrilateral $ABCD$ has sides that are tangent to the circumcircle of $ABCD$ at $A$, $B$, $C$ and $D$.) Prove that the sequence always terminates, except when $\mathcal{Q}_0$ is a square.
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 $PQ=UT$.
Let $ A,B,C$ be collinear points on the plane with $ B$ between $ A$ and $ C$. Equilateral triangles $ ABD,BCE,CAF$ are constructed with $ D,E$ on one side of the line $ AC$ and $ F$ on the other side. Prove that the centroids of the triangles are the vertices of an equilateral triangle, and that the centroid of this triangle lies on the line $ AC$.
Prove that if $n$ is large enough, among any $n$ points of plane we can find $1000$ points such that these $1000$ points have pairwise distinct distances. Can you prove the assertion for $n^{\alpha}$ where $\alpha$ is a positive real number instead of $1000$?
Let $ABC$ be a triangle with $AB \ne AC$. Let the angle bisector of $\angle BAC$ intersects $BC$ at $P$ and intersects the perpendicular bisector of segment $BC$ at $Q$. Prove that $\frac{PQ}{AQ} =\left( \frac{BC}{AB + AC}\right)^2$
Let $\triangle ABC$ be a scalene and acute triangle in which the angle at $A$ is second largest, $H$ is the orthocenter, and $k$ is the circumcircle with center $O$. Let the circumcircle of $\triangle AHO$ intersect the sides $AB$ and $AC$ again at $M$ and $N$, respectively, whereas the altitudes $CH$ and $BH$ intersect $k$ again at $K$ and $L$, respectively. Prove that the intersection of $KL$ and the perpendicular bisector of $AH$ is the orthocenter of $\triangle AMN$.
Proposed by [i]Ilija Jovcevski[/i]
Let $ABC$ be an acute triangle. The circumference with diameter $AB$ intersects sides $AC$ and $BC$ at $E$ and $F$ respectively. The tangent lines to the circumference at the points $E$ and $F$ meet at $P$. Show that $P$ belongs to the altitude from $C$ of triangle $ABC$.
Let $k$ be a circle with radius $r$ and $AB$ a chord of $k$ such that $AB > r$. Furthermore, let $S$ be the point on the chord $AB$ satisfying $AS = r$. The perpendicular bisector of $BS$ intersects $k$ in the points $C$ and $D$. The line through $D$ and $S$ intersects $k$ for a second time in point $E$. Show that the triangle $CSE$ is equilateral.
[i]Proposed by Stefan Leopoldseder[/i]
Two cirlces $ C_1$ and $ C_2$, with center $ O_1$ and $ O_2$ respectively, intersect at $ A$ and $ B$. Let $ O_1$ lies on $ C_2$. A line $ l$ passes through $ O_1$ but does not pass through $ O_2$. Let $ P$ and $ Q$ be the projection of $ A$ and $ B$ respectively on the line $ l$ and let $ M$ be the midpoint of $ \overline{AB}$. Prove that $ MPQ$ is an isoceles triangle.
$\omega$ is the circumcirle of an acute triangle $ABC$. The tangent line passing through $A$ intersects the tangent lines passing through points $B$ and $C$ at points $K$ and $L$, respectively. The line parallel to $AB$ through $K$ and the line parallel to $AC$ through $L$ intersect at point $P$. Prove that $BP=CP$.
(Author: P. Kozhevnikov)
Let $ ABCD$ be a convex quadrilateral, such that
$ \angle CBD\equal{}2\cdot\angle ADB, \angle ABD\equal{}2\cdot\angle CDB$ and $ AB\equal{}CB$.
Prove that quadrilateral $ ABCD$ is a kite.
Let $ABC$ be an acute triangle with $\angle A < \angle B \le \angle C$, and $O$ its circumcenter. The perpendicular bisector of side $AB$ intersects side $AC$ at $D$. The perpendicular bisector of side $AC$ intersects side $AB$ at $E$. Express the angles of triangle $DEO$ in terms of the angles of triangle $ABC$.