Is it possible to place within a square an equilateral triangle whose area is larger than $9/ 20$ of the area of the square?

Let $ABC$ be a triangle. Points $P$ and $Q$ lie on side $BC$ and satisfy $|BP| =|PQ| = |QC| = \frac13 |BC|$. Points $R$ and $S$ lie on side $CA$ and satisfy $|CR| =|RS| = |SA| = 1 3 |CA|$. Finally, points $T$ and $U$ lie on side $AB$ and satisfy $|AT| = |TU| = |UB| =\frac13 |AB|$. Points $P, Q,R, S, T$ and $U$ turn out to lie on a common circle. Prove that $ABC$ is an equilateral triangle.

Nine points, no three of which lie on the same straight line, are located inside an equilateral triangle of side $4$. Prove that some three of these points are vertices of a triangle whose area is not greater than $\sqrt3$.

In the triangle $ABC$ on the sides $AB$ and $AC$ outward constructed equilateral triangles $ABD$ and $ACE$. The segments $CD$ and $BE$ intersect at point $F$. It turns out that point $A$ is the center of the circle inscribed in triangle $ DEF$. Find the angle $BAC$. (Rozhkova Maria)

In a plane there is a triangle $ABC$. Line $AC$ is tangent to circle $c_A$ at point $C$ and circle $c_A$ passes through point $B$. Line $BC$ is tangent to circle $c_B$ at point $C$ and circle $c_B$ passes through point $A$. The second intersection point $S$ of circles $c_A$ and $c_B$ coincides with the incenter of triangle $ABC$. Prove that the triangle $ABC$ is equilateral.

Consider (in the plane) three concentric circles with radii $1, 2$ and $3$ and equilateral triangle $\Delta$ such that on each of the three circles is one vertex of $\Delta$ . Calculate the length of the side of $\Delta$ . [img]https://1.bp.blogspot.com/-q40dl3TSQqE/Xy1QAcno_9I/AAAAAAAAMR8/11nsSA0syNAaGb3W7weTHsNpBeGQZXnHACLcBGAsYHQ/s0/flanders%2B2013%2Bp4.png[/img]

Consider the triangle $ABC$ and the points $D \in (BC),E \in (CA), F \in (AB)$, such that $\frac{BD}{DC}=\frac{CE}{EA}=\frac{AF}{FB}$. Prove that if the circumcenters of triangles $DEF$ and $ABC$ coincide, then the triangle $ABC$ is equilateral.

Given two equilateral triangles $A_1B_1C_1$ and $A_2B_2C_2$ in the plane. (The vertices are mentioned counterclockwise.) We draw vectors $\overrightarrow{OA}, \overrightarrow{OB}, \overrightarrow{OC}$, from the arbitrary point $O$, equal to $\overrightarrow{A_1A_2}, \overrightarrow{B_1B_2}, \overrightarrow{C_1C_2}$ respectively. Prove that the triangle $ABC$ is equilateral.

Given three points $A, B$ and $C$ (not collinear) construct the equilateral triangle of greater perimeter such that each of its sides passes through one of the given points.

Let $ABC$ be an equiangular triangle with circumcircle $\omega$. Let point $F\in AB$ and point $E\in AC$ so that $\angle ABE+\angle ACF=60^{\circ}$. The circumcircle of triangle $AFE$ intersects the circle $\omega$ in the point $D$. The halflines $DE$ and $DF$ intersect the line through $B$ and $C$ in the points $X$ and $Y$. Prove that the incenter of the triangle $DXY$ is independent of the choice of $E$ and $F$. (The angles in the problem statement are not directed. It is assumed that $E$ and $F$ are chosen in such a way that the halflines $DE$ and $DF$ indeed intersect the line through $B$ and $C$.)

The circle divides each side of an equilateral triangle into three equal parts. Prove that the sum of the squares of the distances from any point of this circle to the vertices of the triangle is constant.

Given equilateral triangle $ABC$. Some line, parallel to $[AC]$ crosses $[AB]$ and $[BC]$ in $M$ and $P$ points respectively. Let $D$ be the centre of $PMB$ triangle, $E$ be the midpoint of the $[AP]$ segment. Find the angles of triangle $DEC$ .

Given an equilateral triangle $ABC$ and a point $P$ so that the distances $P$ to $A$ and to $C$ are not farther than the distances $P$ to $B$. Prove that $PB = PA + PC$ if and only if $P$ lies on the circumcircle of $\vartriangle ABC$.

In the plane are given a segment $AC$ and a point $B$ on the segment. Let us draw the positively oriented isosceles triangles $ABS_1, BCS_2$, and $CAS_3$ with the angles at $S_1,S_2,S_3$ equal to $120^o$. Prove that the triangle $S_1S_2S_3$ is equilateral.

The student drew a right triangle $ABC$ on the board with a right angle at the vertex $B$ and inscribed in it an equilateral triangle $KMP$ such that the points $K, M, P$ lie on the sides $AB, BC, AC$, respectively, and $KM \parallel AC$. Then the picture was erased, leaving only points $A, P$ and $C$. Restore erased points and lines.

A circle $C$ with radius $3$ has an equilateral triangle inscribed in it. Let $D$ be a circle lying outside the equilateral triangle, tangent to $C$, and tangent to the equilateral triangle at the midpoint of one of its sides. The radius of $D$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

The triangle $ABC$ is inscribed in the circle. Construct a point $P$ such that the points of intersection of lines $AP, BP$ and $CP$ with this circle are the vertices of an equilateral triangle. (A. Zaslavsky)

On the sides of the parallelogram $ABCD$ outside it are constructed equilateral triangles $ABM$, $BCN$, $CDP$, $ADQ$. Prove that $MNPQ$ is a parallelogram.

Given an equilateral $\Delta ABC$, in which ${{A} _ {1}}, {{B} _ {1}}, {{C} _ {1}}$ are the midpoint of the sides $ BC, \, \, AC, \, \, AB$ respectively. The line $l$ passes through the vertex $A$, we denote by $P, Q$ the projection of the points $B, C$ on the line $l$, respectively (the line $ l $ and the point $Q, \, \, A, \, \, P$ are located as shown in fig.). Denote by $T $ the intersection point of the lines ${{B} _ {1}} P$ and ${{C} _ {1}} Q$. Prove that the line ${{A} _ {1}} T$ is perpendicular to the line $l$. [img]https://cdn.artofproblemsolving.com/attachments/4/b/61f2f4ec9e6b290dfcd47e9351110bebd3bd43.png[/img] (Serdyuk Nazar)

Consider the equilateral triangle $ABC$ with $|BC| = |CA| = |AB| = 1$. On the extension of side $BC$, we define points $A_1$ (on the same side as B) and $A_2$ (on the same side as C) such that $|A_1B| = |BC| = |CA_2| = 1$. Similarly, we define $B_1$ and $B_2$ on the extension of side $CA$ such that $|B_1C| = |CA| =|AB_2| = 1$, and $C_1$ and $C_2$ on the extension of side $AB$ such that $|C_1A| = |AB| = |BC_2| = 1$. Now the circumcentre of 4ABC is also the centre of the circle that passes through the points $A_1,B_2,C_1,A_2,B_1$ and $C_2$. Calculate the radius of the circle through $A_1,B_2,C_1,A_2,B_1$ and $C_2$. [asy] unitsize(1.5 cm); pair[] A, B, C; A[0] = (0,0); B[0] = (1,0); C[0] = dir(60); A[1] = B[0] + dir(-60); A[2] = C[0] + dir(120); B[1] = C[0] + dir(60); B[2] = A[0] + dir(240); C[1] = A[0] + (-1,0); C[2] = B[0] + (1,0); draw(A[1]--A[2]); draw(B[1]--B[2]); draw(C[1]--C[2]); draw(circumcircle(A[2],B[1],C[2])); dot("$A$", A[0], SE); dot("$A_1$", A[1], SE); dot("$A_2$", A[2], NW); dot("$B$", B[0], SW); dot("$B_1$", B[1], NE); dot("$B_2$", B[2], SW); dot("$C$", C[0], N); dot("$C_1$", C[1], W); dot("$C_2$", C[2], E); [/asy]

Given is the equilateral triangle $ABC$ with center $M$. On $CA$ and $CB$ the respective points $D$ and $E$ lie such that $CD = CE$. $F$ is such that $DMFB$ is a parallelogram. Prove that $\vartriangle MEF$ is equilateral.

Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $\overarc {AB}$ . Prove that $AD + BD = DC$.

Consider the isosceles right triangle$ ABC, \angle A = 90^o$, and point $D \in (AB)$ such that $AD = \frac13 AB$. In the half-plane determined by the line $AB$ and point $C$ , consider a point $E$ such that $\angle BDE = 60^o$ and $\angle DBE = 75^o$. Lines $BC$ and $DE$ intersect at point $G$, and the line passing through point $G$ parallel to the line $AC$ intersects the line $BE$ at point $H$. Prove that the triangle $CEH$ is equilateral.

$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle. [img]https://4.bp.blogspot.com/-o0lhdRfRxl0/XNYtJgpJMmI/AAAAAAAAKKg/lmj7KofAJosBZBJcLNH0JKjW3o17CEMkACK4BGAYYCw/s1600/may4_2.gif[/img]

Inside a right angle is given a point $A$. Construct an equilateral triangle, one of the vertices of which is point $A$, and two others lie on the sides of the angle (one on each side).