The line passing through the center of the equilateral triangle $ ABC $ intersects the lines $ AB $, $ BC $ and $ CA $ at the points $ {{C} _ {1}} $, $ {{A} _ {1}} $ and $ {{B} _ {1}} $, respectively. Let $ {{A} _ {2}} $ be a point that is symmetric $ {{A} _ {1}} $ with respect to the midpoint of $ BC $; the points $ {{B} _ {2}} $ and $ {{C} _ {2}} $ are defined similarly. Prove that the points $ {{A} _ {2}} $, $ {{B} _ {2}} $ and $ {{C} _ {2}} $ lie on the same line tangent to the inscribed circle of the triangle $ ABC $. (Serdyuk Nazar)

Let $a$, $b$, $c$ be real numbers, such that $a+b+c=0$ and $a^4+b^4+c^4=50$. Determine the value of $ab+bc+ca$.

In $\vartriangle ABC$ on the sides $BC, CA, AB$ mark feet of altitudes $H_1, H_2, H_3$ and the midpoint of sides $M_1, M_3, M_3$. Let $H$ be orthocenter $\vartriangle ABC$. Suppose that $X_2, X_3$ are points symmetric to $H_1$ wrt $BH_2$ and $CH_3$. Lines $M_3X_2$ and $M_2X_3$ intersect at point $X$. Similarly, $Y_3,Y_1$ are points symmetric to $H_2$ wrt $C_3H$ and $AH_1$.Lines $M_1Y_3$ and $M_3Y_1$ intersect at point $Y.$ Finally, $Z_1,Z_2$ are points symmetric to $H_3$ wrt $AH_1$ and $BH_2$. Lines $M_1Z_2$ and $M_2Z_1$ intersect at the point $Z$ Prove that $H$ is the incenter $\vartriangle XYZ$ .

Let $ABC$ be a triangle with $\angle BAC =60^\circ$. $A'$ is the symmetric point of $A$ wrt $\overline{BC}$. $D$ is the point in $\overline{AC}$ such that $\overline{AB}=\overline{AD}$. $H$ is the orthocenter of triangle $ABC$. $l$ is the external angle bisector of $\angle BAC$. $\{M\}=\overline{A'D}\cap l$,$\{N\}=\overline{CH} \cap l$. Show that $\overline{AM}=\overline{AN}$.

Inside the circle are given three points that do not belong to one line. In one step it is allowed to replace one of the points with a symmetric one wrt the line containing the other two points. Is it always possible for a finite number of these steps to ensure that all three points are outside the circle?

Let $[ABC]$ be any triangle and let $D, E$ and $F$ be the symmetrics of the circumcenter wrt the three sides. Prove that the triangles $[ABC]$ and $[DEF]$ are congruent. [img]https://cdn.artofproblemsolving.com/attachments/c/6/45bd929dfff87fb8deb09eddb59ef46e0dc0f4.png[/img]

Inside the convex quadrilateral $ABCD$ lies the point $'M$. Reflect it symmetrically with respect to the midpoints of the sides of the quadrilateral and connect the obtained points so that they form a convex quadrilateral. Prove that the area of ​​this quadrilateral does not depend on the choice of the point $M$.

Given an acute triangle $ABC$, $(c)$ its circumcircle with center $O$ and $H$ the orthocenter of the triangle $ABC$. The line $AO$ intersects $(c)$ at the point $D$. Let $D_1, D_2$ and $H_2, H_3$ be the symmetrical points of the points $D$ and $H$ with respect to the lines $AB, AC$ respectively. Let $(c_1)$ be the circumcircle of the triangle $AD_1D_2$. Suppose that the line $AH$ intersects again $(c_1)$ at the point $U$, the line $H_2H_3$ intersects the segment $D_1D_2$ at the point $K_1$ and the line $DH_3$ intersects the segment $UD_2$ at the point $L_1$. Prove that one of the intersection points of the circumcircles of the triangles $D_1K_1H_2$ and $UDL_1$ lies on the line $K_1L_1$.

Let $ABC$ be a triangle and $D$ a point on the side $BC$. Point $E$ is the symmetric of $D$ with respect to $AB$. Point $F$ is the symmetric of $E$ with respect to $AC$. Point $P$ is the intersection of line $DF$ with line $AC$. Prove that the quadrilateral $AEDP$ is cyclic. (Malik Talbi)

Let $ABCDA' B' C' D '$ be a cube , $M$ the foot of the perpendicular from $A$ on the plane $(A'CD)$, $N$ the foot of the perpendicular from $B$ on the diagonal $A'C$ and $P$ is symmetric of the point $D$ with respect to $C$. Show that the points $M, N, P$ are collinear.

Given triangle $ABC$. Points $A_1,B_1$ and $C_1$ are symmetric to its vertices with respect to opposite sides. $C_2$ is the intersection point of lines $AB_1$ and $BA_1$. Points$ A_2$ and $B_2$ are defined similarly. Prove that the lines $A_1 A_2, B_1 B_2$ and $C_1 C_2$ are parallel. (A. Zaslavsky)

Let $ABC$ be a triangle, $I_a$ the center of the excircle at side $BC$, and $M$ its reflection across $BC$. Prove that $AM$ is parallel to the Euler line of the triangle $BCI_a$.

Let $ABC$ be a triangle. Let $B'$ be the symmetric of $B$ with respect to $C, C'$ the symmetry of $C$ with respect to $A$ and $A'$ the symmetry of $A$ with respect to $B$. a) Prove that the area of triangle $AC'A'$ is twice the area of triangle $ABC$. b) If we delete points $A, B, C$, how can they be reconstituted? Justify your reasoning.

On the plane, there is drawn a parallelogram $P$ and a point $X$ outside of $P$. Using only an ungraded rule, determine the point $W$ that is symmetric to $X$ with respect to the center $O$ of $P$.

It is known that the graph of the function $y =\frac{a-6x}{2+x}$ is centrally summetric to the graph of the function $y = \frac{1}{x}$ with respect to some point. Find the value of the parameter $a$ and the coordinates of the center of symmetry.

An acute triangle $ABC$ is given in which $AB <AC$. Point $E$ lies on the $AC$ side of the triangle, with $AB = AE$. The segment $AD$ is the diameter of the circumcircle of the triangle $ABC$, and point $S$ is the center of this arc $BC$ of this circle to which point $A$ does not belong. Point $F$ is symmetric of point $D$ wrt $S$. Prove that lines $F E$ and $AC$ are perpendicular.