Let $ABC$ be a triagle inscribed in a circle $(O)$. A variable line through the orthocenter $H$ of the triangle meets the circle $(O)$ at two points $P , Q$. Two lines through $P, Q$ that are perpendicular to $AP , AQ$ respectively meet $BC$ at $M, N$ respectively. Prove that the line through $P$ perpendicular to $OM$ and the line through $Q$ perpendicular to $ON$ meet each other at a point on the circle $(O)$. Nguyễn Văn Linh

Let $ABC$ be an isosceles triangle with $AB=AC$ and let $\Gamma$ denote its circumcircle. A point $D$ is on arc $AB$ of $\Gamma$ not containing $C$. A point $E$ is on arc $AC$ of $\Gamma$ not containing $B$. If $AD=CE$ prove that $BE$ is parallel to $AD$.

Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.

Let points $A$, $B$, $C$ lie on a circle, and line $b$ be the tangent to the circle at point $B$. Perpendiculars $PA_1$ and $PC_1$ are dropped from a point $P$ on line $b$ onto lines $AB$ and $BC$ respectively. Points $A_1$ and $C_1$ lie inside line segments $AB$ and $BC$ respectively. Prove that $A_1C_1$ is perpendicular to $AC$.

Point $O$ is the intersection point of the heights of an acute triangle $\vartriangle ABC$. Prove that the three circles which pass: a) through $O, A, B$, b) through $O, B, C$, and c) through $O, C, A$, are equal

Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.

A circle of radius $2$ is inscribed in equilateral triangle $ABC$. The altitude from $A$ to $BC$ intersects the circle at a point $D$ not on $BC$. $BD$ intersects the circle at a point $E$ distinct from $D$. Find the length of $BE$.

Equilateral triangle $ T$ is inscribed in circle $ A$, which has radius $ 10$. Circle $ B$ with radius $ 3$ is internally tangent to circle $ A$ at one vertex of $ T$. Circles $ C$ and $ D$, both with radius $ 2$, are internally tangent to circle $ A$ at the other two vertices of $ T$. Circles $ B$, $ C$, and $ D$ are all externally tangent to circle $ E$, which has radius $ \frac {m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$. [asy]unitsize(2.2mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90); pair Ep=(0,4-27/5); pair[] dotted={A,B,C,D,Ep}; draw(Circle(A,10)); draw(Circle(B,3)); draw(Circle(C,2)); draw(Circle(D,2)); draw(Circle(Ep,27/5)); dot(dotted); label("$E$",Ep,E); label("$A$",A,W); label("$B$",B,W); label("$C$",C,W); label("$D$",D,E);[/asy]

In the acute triangle $ABC$, $\angle BAC$ is less than $\angle ACB $. Let $AD$ be a diameter of $\omega$, the circle circumscribed to said triangle. Let $E$ be the point of intersection of the ray $AC$ and the tangent to $\omega$ passing through $B$. The perpendicular to $AD$ that passes through $E$ intersects the circle circumscribed to the triangle $BCE$, again, at the point $F$. Show that $CD$ is an angle bisector of $\angle BCF$.

Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.

Let $ABCD$ be cyclic quadrilateral,let $\gamma$ be it's circumscribed circle and let $M$ be the midpoint of arc $AB$ of $\gamma$,which does not contain points $C,D$.The line that passes through $M$ and the intersection point of diagonals $AC,BD$,intersects $\gamma$ in $N\neq M$. Let $P,Q$ be two points situated on $CD$,such that $\angle{AQD}=\angle{DAP}$ and $\angle{BPC}=\angle{CBQ}$.Prove that circles $\odot(NPQ)$ and $\gamma$ are tangent.

let a triangle ABC and A',B',C' be the midpoints of the arcs BC,CA,AB respectively of its circumcircle. A'B',A'C' meets BC at $ A_1,A_2$ respectively. Pairs of point $ (B_1,B_2),(C_1,C_2)$ are similarly defined. Prove that $ A_1A_2 \equal{} B_1B_2 \equal{} C_1C_2$ if and only if triangle ABC is equilateral.

Let $ABCD$ be a cyclic quadrilateral with circumcentre $O,$ and the pair of opposite sides not parallel with each other. Let $M=AB\cap CD$ and $N=AD\cap BC.$ Denote, by $P,Q,S,T;$ the intersection of the internal angle bisectors of $\angle MAN$ and $\angle MBN;$ $\angle MBN$ and $\angle MCN;$ $\angle MDN$ and $\angle MAN;$ $\angle MCN$ and $\angle MDN.$ Suppose that the four points $P,Q,S,T$ are distinct. (a) Show that the four points $P,Q,S,T$ are concyclic. Find the centre of this circle, and denote it as $I.$ (b) Let $E=AC\cap BD.$ Prove that $E,O,I$ are collinear.

Let $ABC$ be an acute-angled triangle, and let $F$ be the foot of its altitude from the vertex $C$. Let $M$ be the midpoint of the segment $CA$. Assume that $CF=BM$. Then the angle $MBC$ is equal to angle $FCA$ if and only if the triangle $ABC$ is equilateral.

If the inradius of a triangle is half of its circumradius, prove that the triangle is equilateral.

In a triangle $ABC$, let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$. The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$, and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$. Prove that $H$ is the incentre of triangle $PQR$.

Given an acute triangle $ABC$, let $D$, $E$ and $F$ be points in the lines $BC$, $AC$ and $AB$ respectively. If the lines $AD$, $BE$ and $CF$ pass through $O$ the centre of the circumcircle of the triangle $ABC$, whose radius is $R$, show that: \[\frac{1}{AD}\plus{}\frac{1}{BE}\plus{}\frac{1}{CF}\equal{}\frac{2}{R}\]

Let $D$, $E$, $F$ be points on the sides $BC$, $CA$, $AB$ respectively of a triangle $ABC$ (distinct from the vertices). If the quadrilateral $AFDE$ is cyclic, prove that \[ \frac{ 4 \mathcal A[DEF] }{\mathcal A[ABC] } \leq \left( \frac{EF}{AD} \right)^2 . \] [i]Greece[/i]

Let $ABC$ be an acute triangle and $O$ be its circumcenter. Let $D$ be the midpoint of $[AB]$. The circumcircle of $\triangle ADO$ meets $[AC]$ at $A$ and $E$. If $|AE|=7$, $|DE|=8$, and $m(\widehat{AOD}) = 45^\circ$, what is the area of $\triangle ABC$? $ \textbf{(A)}\ 56\sqrt 3 \qquad\textbf{(B)}\ 56 \sqrt 2 \qquad\textbf{(C)}\ 50 \sqrt 2 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ \text{None of the preceding} $

In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$. Prove that $\angle A = 2\angle B$ if and only if $AC = 2 MD$.

Let $A,B$ be two fixed points on a plane and $\Omega$ a fixed semicircle arc with diameter $AB$. Let $T$ be another fixed point on $\Omega$, and $\omega$ a fixed circle that passes through $A$ and $T$ and has its center in $\Delta ABT$. Let $P$ be a moving point on the arc $TB$ (endpoints excluded), and $C,D$ be two moving points on $\omega$ such that $C$ lies on segment $AP$, $C,D$ lies on different sides of line $AB$ and $CD\ \bot \ AB$. Denote the circumcenter of $\Delta CDP$ of $K$. Prove that (i) $K$ lies on the circumcircle of $\Delta TDP$. (ii) $K$ is a fixed point.

3 segments $AA_1$, $BB_1$, $CC_1$ in space share a common midpoint $M$. Turns out, the sphere circumscribed about the tetrahedron $MA_1B_1C_1$ is tangent to plane $ABC$ at point $D$. Point $O$ is the circumcenter of triangle $ABC$. Prove that $MO = MD$.

Let $O$ be a circumcenter of acute triangle $ABC$ and let $O_1$ and $O_2$ be circumcenters of triangles $OAB$ and $OAC$, respectively. Circumcircles of triangles $OAB$ and $OAC$ intersect side $BC$ in points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Perpendicular bisector of side $BC$ intersects side $AC$ in point $F$($F \neq A$). Prove that circumcenter of triangle $ADE$ lies on $AC$ iff $F$ lies on line $O_1O_2$

Let $\Gamma$ be the circumcircle of isosceles triangle $ABC$ with vertex $C$. An arbitrary point $M$ is chosen on the segment $BC$ and point $N$ lies on the ray $AM$ with $M$ between $A,N$ such that $AN=AC$. The circumcircle of $CMN$ cuts $\Gamma$ in $P$ other than $C$ and $AB,CP$ intersect at $Q$. Prove that $\angle BMQ = \angle QMN.$

Let $ C_1$ be a circle and $ P$ be a fixed point outside the circle $ C_1$. Quadrilateral $ ABCD$ lies on the circle $ C_1$ such that rays $ AB$ and $ CD$ intersect at $ P$. Let $ E$ be the intersection of $ AC$ and $ BD$. (a) Prove that the circumcircle of triangle $ ADE$ and the circumcircle of triangle $ BEC$ pass through a fixed point. (b) Find the the locus of point $ E$.