Found problems: 296
Let $ABC$ be an equilateral triangle . Let point $D$ lie on side $AB,E$ lie on side $AC, D_1$ and $E_1$ lie on side BC such that $AB=DB+BD_1$ and $AC=CE+CE_1$. Calculate the smallest angle between the lines $DE_1$ and $ED_1$.
Let $ABC$ be an equilateral triangle with side length $10$. A square $PQRS$ is inscribed in it, with $P$ on $AB, Q, R$ on $BC$ and $S$ on $AC$. If the area of the square $PQRS$ is $m +n\sqrt{k}$ where $m, n$ are integers and $k$ is a prime number then determine the value of $\sqrt{\frac{m+n}{k^2}}$.
Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.
The circle inscribed in the triangle $ ABC $ , with center $O$, touches the sides $ AB $ and $ BC $ at the points $ C_1 $ and $ A_1 $, respectively. The lines $ CO $ and $ AO $ intersect the line $ C_1A_1 $ at the points $ K $ and $ L $. $ M $ is the midpoint of $ AC $ and $ \angle ABC = 60^\circ $. Prove that $ KLM $ is a regular 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 $\sqrt3$.
A line $g$ and a point $A$ outside $g$ are given in a plane. A point $P$ moves along $g$.
Find the locus of the third vertices of equilateral triangles whose two vertices are $A$ and $P$.
In the parallelogram $ABCD$, all sides are equal, and $\angle A = 60^o$. Let $F$ be a point on line $AD, H$ a point on line $DC$, and $G$ a point on diagonal $AC$ such that $DFGH$ is a parallelogram. Show that then $\vartriangle BHF$ 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]
Let $ABC$ be an equilateral triangle. The points $M$ and $K$ are located in different half-planes with respect to line $BC$, so that the point $M \in (AB)$ ¸and the triangle $MKC$ is equilateral. Prove that the lines $AC$ and $BK$ are parallel.
Given a straight line with points $A, B, C$ and $D$. Construct using $AB$ and $CD$ regular triangles (in the same half-plane). Let $E,F$ be the third vertex of the two triangles (as in the figure) . The circumscribed circles of triangles $AEC$ and $BFD$ intersect in $G$ ($G$ is is in the half plane of triangles). Prove that the angle $AGD$ is $120^o$
[img]https://1.bp.blogspot.com/-66akc83KSs0/X9j2BBOwacI/AAAAAAAAM0M/4Op-hrlZ-VQRCrU8Z3Kc3UCO7iTjv5ZQACLcBGAsYHQ/s0/2011%2BDurer%2BC5.png[/img]
An equilateral triangle of altitude $1$ is considered. For every point $P$ on the interior of the triangle, denote by $x, y , z$ the distances from the point $P$ to the sides of the triangle.
a) Prove that for every point $P$ inside the triangle it is true that $x + y + z = 1$.
b) For which points of the triangle does it hold that the distance to one side is greater than the sum of the distances to the other two?
c) We have a bar of length $1$ and we break it into three pieces. find the probability that with these pieces a triangle can be formed.
Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$. Find the measure of the angle $\angle PBC$ .
Three equilateral triangles $ABC, CDE, EHK$ (the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors $\overrightarrow{AD}$ and $\overrightarrow{DK}$ are equal. Prove that the triangle $BHD$ is also equilateral
In an equilateral triangle $ ABC $, point $ O $ is chosen and perpendiculars $ OM $, $ ON $, $ OP $ are dropped to the sides $ BC $, $ CA $, $ AB $, respectively. Prove that the sum of the segments $ AP $, $ BM $, $ CN $ does not depend on the position of point $ O $.
A set of points in the plane is called [i]free [/i] if no three points of the set are the vertices of an equilateral triangle. Prove that any set of $n$ points in the plane has a free subset of at least $\sqrt{n}$ points
In an equilateral triangle $ABC$, whose side is $4$, the line perpendicular to $AB$ is drawn through the point $ A$, the line perpendicular to $BC$ through point $ B$ and the line perpendicular to $CA$ through point $C$. These three lines determine another triangle. Calculate the perimeter of this triangle
In a triangle $ABC$ with medians $AD$ and $BE$ , holds that $\angle CAD= \angle CBE=30^o$. Prove that triangle $ABC$ is equilateral.
A point $M$ is found inside a convex quadrilateral $ABCD$ such that triangles $AMB$ and $CMD$ are isoceles ($AM = MB, CM = MD$) and $\angle AMB= \angle CMD = 120^o$ . Prove that there exists a point N such that triangles$ BNC$ and $DNA$ are equilateral.
(I.Sharygin)
An arbitrary point $M$ inside an equilateral triangle $ABC$ was connected to vertices. Prove that on each side the triangle can be selected one point at a time so that the distances between them would be equal to $AM, BM, CM$.
We have $10,000$ identical equilateral triangles. Consider the largest regular hexagon that can be formed with these triangles without overlapping. How many triangles will not be used?
The shape of a dog kennel from above is an equilateral triangle with side length $1$ m and its corners in points $A, B$ and $C$, as shown in the picture. The chain of the dog is of length $6$ m and its end is fixed to the corner in point $A$. The dog himself is in point $K$ in a way that the chain is tight and points $K, A$ and $B$ are on the same straight line. The dog starts to move clockwise around the kennel, holding the chain tight all the time. How long is the walk of the dog until the moment when the chain is tied round the kennel at full?
[img]https://cdn.artofproblemsolving.com/attachments/9/5/616f8adfe66e2eb60f1a6c3f26e652c45f3e27.png[/img]
Equilateral triangle $ABC$ has side length $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the “lune” that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p}-\frac{q\pi}{r}$, where $p, q$, and $ r$ are positive integers such that $q$ and $r$ are relatively prime. Compute $p + q + r$.
[img]https://cdn.artofproblemsolving.com/attachments/7/7/f349a807583a83f93ba413bebf07e013265551.png[/img]
Points $A, B$, and $C$ are on a line in this order. Points $D$ and $E$ lie on the same side of this line, in such a way that triangles $ABD$ and $BCE$ are equilateral. The segments $AE$ and $CD$ intersect in point $S$. Prove that $\angle ASD = 60^o$.
[asy]
unitsize(1.5 cm);
pair A, B, C, D, E, S;
A = (0,0);
B = (1,0);
C = (2.5,0);
D = dir(60);
E = B + 1.5*dir(60);
S = extension(C,D,A,E);
fill(A--B--D--cycle, gray(0.8));
fill(B--C--E--cycle, gray(0.8));
draw(interp(A,C,-0.1)--interp(A,C,1.1));
draw(A--D--B--E--C);
draw(A--E);
draw(C--D);
draw(anglemark(D,S,A,5));
dot("$A$", A, dir(270));
dot("$B$", B, dir(270));
dot("$C$", C, dir(270));
dot("$D$", D, N);
dot("$E$", E, N);
dot("$S$", S, N);
[/asy]
Consider the equilateral triangle $ABC$ with center of gravity $G$. Let $M$ be a point, inside the triangle and $O$ be the midpoint of the segment $MG$. Three segments go through $M$, each parallel to one side of the triangle and with the ends on the other two sides of the given triangle.
a) Show that $O$ is at equal distances from the midpoints of the three segments considered.
b) Show that the midpoints of the three segments are the vertices of an equilateral triangle.
There is a point inside an equilateral triangle side $d$ whose distance from the vertices is $3, 4, 5$. Find $d$.