The board features a triangle $ABC$, its center of the circle circumscribed is the point $O$, the midpoint of the side $BC$ is the point $F$, and also some point $K$ on side $AC$ (see fig.). Master knowing that $\angle BAC$ of this triangle is equal to the sharp angle $\alpha$ has separately drawn an angle equal to $\alpha$. After this teacher wiped the board, leaving only the points $O, F, K$ and the angle $\alpha$. Is it possible with a compass and a ruler to construct the triangle $ABC$ ? Justify the answer. (Grigory Filippovsky) [img]https://1.bp.blogspot.com/-RRPt8HbqW4I/XObthZFXyyI/AAAAAAAAKOo/zfHemPjUsI4XAfV_tcmKA6_al0i_gQ9iACK4BGAYYCw/s1600/Yasinsky%2B2019%2Bp6.png[/img]

Let $I$ and $O$ be respectively the incentre and circumcentre of a triangle $ABC$. If $AB = 2$, $AC = 3$ and $\angle AIO = 90^{\circ}$, find the area of $\triangle ABC$.

Points $E, F$ are in the plane of triangle $ABC$ so that triangles $ABE$ and $ACF$ are the opposite directed, and the two triangles are isosceles in that $BE = AE, AF = CF$. Let $H, K$ be the orthocenter of triangle $ABE, ACF$ respectively. Points $M, N$ are the intersections of $BE$ and $CF, CK$ and $CH$. Prove that $MN$ passes through the center of the circumcircle of triangle $ABC$. Nguyen Minh Ha, High School for Education, Hanoi Pedagogical University

Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be the intersection of straight lines $BO$ and $AC$ and $\omega$ be the circumcircle of triangle $AOP$. Suppose that $BO = AP$ and that the measure of the arc $OP$ in $\omega$, that does not contain $A$, is $40^o$. Determine the measure of the angle $\angle OBC$. [img]https://3.bp.blogspot.com/-h3UVt-yrJ6A/XqBItXzT70I/AAAAAAAAL2Q/7LVv0gmQWbo1_3rn906fTn6wosY1-nIfwCK4BGAYYCw/s1600/2007%2Bomb%2Bl2.png[/img]

Let $ABC$ be an acute, non-isosceles triangle which is inscribed in a circle $(O)$. A point $I$ belongs to the segment $BC$. Denote by $H$ and $K$ the projections of $I$ on $AB$ and $AC$, respectively. Suppose that the line $HK $ intersects $(O)$ at $M, N$ ($H$ is between $M, K$ and $K$ is between $H, N$). Let $X, Y$ be the centers of the circles $(ABK),(ACH)$ respectively. Prove the following assertions: a) If $I$ is the projection of $A$ on $BC$, then $A$ is the center of circle $(IMN)$. b) If $XY\parallel BC$, then the orthocenter of $XOY$ is the midpoint of $IO$.

Let $AC$ be the hypothenuse of a right-angled triangle $ABC$. The bisector $BD$ is given, and the midpoints $E$ and $F$ of the arcs $BD$ of the circumcircles of triangles $ADB$ and $CDB$ respectively are marked (the circles are erased). Construct the centers of these circles using only a ruler.

Let $ABC$ be an acute-angled triangle with $\angle ACB = 60^o$ , and its heights $AD$ and $BE$ intersect at point $H$. Prove that the circumcenter of triangle $ABC$ lies on a line bisecting the angles $AHE$ and $BHD$.

Around triangle $ABC$ with acute angle C is circumscribed a circle. On the arc $AB$, which does not contain point $C$, point $D$ is chosen. Point $D'$ is symmetric on point $D$ with respect to line $AB$. Straight lines $AD'$ and $BD'$ intersect segments $BC$ and $AC$ at points $E$ and $F$. Let point $C$ move along its arc $AB$. Prove that the center of the circumscribed circle of a triangle $CEF$ moves on a straight line.

On sides $AB$ and $AC$ of an acute triangle $\Delta ABC$, with orthocenter $H$ and circumcenter $O$, are given points $P$ and $Q$ respectively such that $APHQ$ is a parallelogram. Prove the following equality: \begin{align*} \frac{PB\cdot PQ}{QA\cdot QO}=2 \end{align*}

A circle with the center $I$ is entirely inside of a circle with center $O$. Consider all possible chords $AB$ of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle $AIB$.

Point $O$ is the intersection of the perpendicular bisectors of the sides of the triangle $\vartriangle ABC$ . Let $D$ be the intersection of the line $AO$ with the segment $[BC]$. Knowing that $OD = BD = \frac 13 BC$, find the measures of the angles of the triangle $\vartriangle ABC$.

Let $O$, $I$ be the circumcenter and the incenter of triangle $ABC$, respectively, and let the incircle tangents $BC$ at $D$. Furthermore, suppose that $H$ is the orthocenter of triangle $BIC$, $N$ is the midpoint of the arc $BAC$, and $X$ is the intersection of $OI$ and $NH$. If $P$ is the reflection of $A$ with respect to $OI$, show that $\odot(IDP)$ and $\odot(IHX)$ are tangent to each other.

A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$. (Karl Czakler)

In triangle $ABC$, $M$ is the intersection point of the medians, $O$ is the center of the inscribed circle. Prove that if the line $OM$ is parallel to the side $BC$, then the point $O$ is equidistant from the sides $AB$ and $AC$.

Let $ABC$ be a triangle with $AB : AC : BC =5:5:6$. Denote by $M$ the midpoint of $BC$ and by $N$ the point on the segment $BC$ such that $BN = 5 \cdot CN$. Prove that the circumcenter of triangle $ABN$ is the midpoint of the segment connecting the incenters of triangles $ABC$ and $ABM$. Slovakia

Let $ABC$ be a triangle, let $O$ be its circumcenter, let $A'$ be the orthogonal projection of $A$ on the line $BC$, and let $X$ be a point on the open ray $AA'$ emanating from $A$. The internal bisectrix of the angle $BAC$ meets the circumcircle of $ABC$ again at $D$. Let $M$ be the midpoint of the segment $DX$. The line through $O$ and parallel to the line $AD$ meets the line $DX$ at $N$. Prove that the angles $BAM$ and $CAN$ are equal.

Let $ABC$ be an acute, non isosceles triangle with $O,H$ are circumcenter and orthocenter, respectively. Prove that the nine-point circles of $AHO,BHO,CHO$ has two common points.

Let $ABC$ be a triangle with $A$ is an obtuse angle. Denote $BE$ as the internal angle bisector of triangle $ABC$ with $E \in AC$ and suppose that $\angle AEB = 45^o$. The altitude $AD$ of triangle $ABC$ intersects $BE$ at $F$. Let $O_1, O_2$ be the circumcenter of triangles $FED, EDC$. Suppose that $EO_1, EO_2$ meet $BC$ at $G, H$ respectively. Prove that $\frac{GH}{GB}= \tan \frac{a}{2}$

Suppose that $A_1A_2A_3$ is a triangle with $A_1A_2 = 16$ and $A_1A_3 = A_2A_3 = 10$. For each integer $n \ge 4$, set An to be the circumcenter of triangle $A_{n-1}A_{n-2}A_{n-3}$. There exists a unique point $Z$ lying in the interiors of the circumcircles of triangles $A_kA_{k+1}A_{k+2}$ for all integers $k \ge 1$. If $ZA^2_1+ ZA^2_2+ ZA^2_3+ ZA^2_4$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$.

Consider an isosceles triangle $ABC$ with $AB = AC$. Let $\omega(XYZ)$ be the circumcircle of the triangle $XY Z$. The tangents to $\omega(ABC)$ through $B$ and $C$ meet at the point $D$. The point $F$ is marked on the arc $AB$ of $\omega(ABC)$ that does not contain $C$. Let $K$ be the point of intersection of lines $AF$ and $BD$ and $L$ the point of intersection of the lines $AB$ and $CF$. Let $T$ and $S$ be the centers of the circles $\omega(BLC)$ and $\omega(BLK)$, respectively. Suppose that the circles $\omega(BTS)$ and $\omega(CFK)$ are tangent to each other at the point $P$. Prove that $P$ belongs to the line $AB$.

Let P be an arbitrary point on the arc $AC$ of the circumcircle of a fixed triangle $ABC$, not containing $B$. The bisector of angle $APB$ meets the bisector of angle $BAC$ at point $P_a$ the bisector of angle $CPB$ meets the bisector of angle $BCA$ at point $P_c$. Prove that for all points $P$, the circumcenters of triangles $PP_aP_c$ are collinear. by I. Dmitriev

Let $\omega$ be a circle in the plane and $A,B$ two points lying on it. We denote by $M$ the midpoint of $AB$ and let $P \ne M$ be a new point on $AB$. Build circles $\gamma$ and $\delta$ tangent to $AB$ at $P$ and to $\omega$ at $C$, respectively $D$. Consider $E$ to be the point diametrically opposed to $D$ in $\omega$. Prove that the circumcenter of $\triangle BMC$ lies on the line $BE$.

There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles: a) points of intersection of heights, b) the intersection points of the medians, c) the centers of the circumscribed circles.

Let $B$ be a point on a circle $k_1, A \ne B$ be a point on the tangent to the circle at $B$, and $C$ a point not lying on $k_1$ for which the segment $AC$ meets $k_1$ at two distinct points. Circle $k_2$ is tangent to line $AC$ at $C$ and to $k_1$ at point $D$, and does not lie in the same half-plane as $B$. Prove that the circumcenter of triangle $BCD$ lies on the circumcircle of $\vartriangle ABC$

Let $O$ be the center of the circumcircle, and $AD$ be the angle bisector of the acute triangle $ABC$. The perpendicular drawn from point $D$ on the line $AO$ ​​intersects the line $AC$ at the point $P$. Prove that $AP = AB$.