Found problems: 296
Equilateral triangles $ABC_1, BCA_1, CAB_1$ are built on the sides of the triangle $ABC$ to the outside. On the segment $A_1B_1$ to the outer side of the triangle $A_1B_1C_1$, an equilateral triangle $A_1B_1C_2$ is constructed. Prove that $C$ is the midpoint of the segment $C_1C_2$.
(A. Zaslavsky)
In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?
Consider two equilateral triangles $ABC$ and $MNP$ with the property that $AB \parallel MN, BC \parallel NP$ and $CA \parallel PM$ , so that the surfaces of the triangles intersect after a convex hexagon. The distances between the three pairs of parallel lines are at most equal to $1$. Show that at least one of the two triangles has the side at most equal to $\sqrt {3}$ .
$ABCD$ 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.
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The target is a triangle divided by three families of parallel lines into $100$ equal regular triangles with single sides. A sniper shoots at a target. He aims at triangle and hits either it or one of the sides adjacent to it. He sees the results of his shooting and can choose when stop shooting. What is the greatest number of triangles he can with a guarantee of hitting five times?
The hexagon $ABCDEF$ is inscribed in a circle. The sides $AB$, $CD$ and $EF$ are all equal in length to the radius. Prove that the midpoints of the other three sides determine an equilateral triangle.
The map of a town shows a plane divided into equal equilateral triangles. The sides of these triangles are streets and their vertices are intersections; $6$ streets meet at each junction. Two cars start simultaneously in the same direction and at the same speed from points $A$ and $B$ situated on the same street (the same side of a triangle). After any intersection an admissible route for each car is either to proceed in its initial direction or turn through $120^o$ to the right or to the left. Can these cars meet? (Either prove that these cars won’t meet or describe a route by which they will meet.)
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A given equilateral triangle of side $10$ is divided into $100$ equilateral triangles of side $1$ by drawing parallel lines to the sides of the original triangle. Find the number of equilateral triangles, having vertices in the intersection points of parallel lines whose sides lie on the parallel lines.
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 $AD + BD = DC$.
Given a segment $AB$ in the plane, choose on it a point $M$ different from $A$ and $B$. Two equilateral triangles $\triangle AMC$ and $\triangle BMD$ in the plane are constructed on the same side of segment $AB$. The circumcircles of the two triangles intersect in point $M$ and another point $N$. (The [b]circumcircle[/b] of a triangle is the circle that passes through all three of its vertices.)
(a) Prove that lines $AD$ and $BC$ pass through point $N$.
(b) Prove that no matter where one chooses the point $M$ along segment $AB$, all lines $MN$ will pass through some fixed point $K$ in the plane.
Find the side of the smallest regular triangle that can be inscribed in a right triangle with an acute angle of $30^o$ and a hypotenuse of $2$. (All vertices of the required regular triangle must be located on different sides of this right triangle.)
Let $AH$ be an altitude of an equilateral triangle $ABC$. Let $I$ be the incentre of triangle $ABH$, and let $L, K$ and $J$ be the incentres of triangles $ABI,BCI$ and $CAI$ respectively. Determine $\angle KJL$.
Consider an equilateral triangle and square, both with area $1$. What is the product of their perimeters?
To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.
Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$. Determine the area of $\vartriangle DEF$.
Let $A_1,B_1,C_1$ be the midpoints of the sides of the $BC,AC, AB$ of an equilateral triangle $ABC$. Around the triangle $A_1B_1C_1$ is a circle $\gamma$, to which the tangents $B_2C_2$, $A_2C_2$, $A_2B_2$ are drawn, respectively, parallel to the sides $BC, AC, AB$. These tangents have no points in common with the interior of triangle $ABC$. Find out the mutual location of the points of intersection of the lines $AA_2$ and $BB_2$, $AA_2$ and $CC_2$, $BB_2$ and $CC_2$ and the circumscribed circle $\gamma$. Try to consider the case of arbitrary points $A_1,B_1,C_1$ located on the sides of the triangle $ABC$.
An equilateral triangle is partitioned into smaller equilateral triangular pieces. Prove that two of the pieces are the same size.
Three different points $A,B$ and $C$ lie on a circle with center $M$ so that $| AB | = | BC |$. Point $D$ is inside the circle in such a way that $\vartriangle BCD$ is equilateral. Let $F$ be the second intersection of $AD$ with the circle . Prove that $| F D | = | FM |$.
$4$ equilateral triangles of side length $1$ are drawn on the interior of a unit square, each one of which shares a side with one of the $4$ sides of the unit square. What is the common area enclosed by all $4$ equilateral triangles?
Cut each of the equilateral triangles with sides $2$ and $3$ into three parts and construct an equilateral triangle from all received parts.
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.
Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into $10$ equal segments and the triangle into $100$ congruent triangles. Each of these $100$ triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”. What is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe?
(R Zhenodarov)
Let $I$ be the incenter of an acute riangle $ABC$. Let $C_A(A, AI)$ be the circle with center $A$ and radius $AI$. Circles $C_B(B, BI)$, $C_C(C, CI) $ are defined in an analogous way. Let $X, Y, Z$ be the intersection points of $C_B$ with $C_C$, $C_C$ with $C_A$, $C_A$ with $C_B$ respectively (different than $I$) . Show that if the radius of the circle that passes through the points $X, Y, Z$ is equal to the radius of the circle that passes through points $A$, $B$ and $C$ then triangle $ABC$ is equilateral.
$A, B, C, D$ are four distinct points such that triangles $ABC$ and $CBD$ are both equilateral. Find as many circles as you can, which are equidistant from the four points. How can these circles be constructed?
[i]Remark: The distance between a point $P$ and a circle c is measured as follows: we join $P$ and the centre of the circle with a straight line, and measure how much we need to travel along thisline (starting from $P$) to hit the perimeter of the circle. If $P$ is an internal point of the circle, the distance is the length of the shorter such segment. The distance between a circle and itscentre is the radius of the circle.[/i]
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 $C$.)