Found problems: 296
Suppose that an interior point $O$ of a triangle $ABC$ is such that the angles $\angle BAO,\angle CBO, \angle ACO$ are all greater than or equal to $30^o$. Prove that the triangle $ABC$ is equilateral.
Let $ A_1,B_1,C_1 $ be points on the sides (excluding their endpoints) $ BC,CA,AB, $ respectively, of a triangle $ ABC, $ such that $ \angle A_1AB =\angle B_1BC=\angle C_1CA. $ Let $ A^* $ be the intersection of $ BB_1 $ with $ CC_1,B^* $ be the intersection of $ CC_1 $ with $ AA_1, $ and $ C^* $ be the intersection of $ AA_1 $ with $ BB_1. $ Denote with $ r_A,r_B,r_C $ the inradii of $ A^*BC,AB^*C,ABC^*, $ respectively. Prove that
$$ \frac{r_A}{BC}=\frac{r_B}{CA}=\frac{r_C}{AB} $$
if and only if $ ABC $ is equilateral.
[i]Daniel Văcărețu[/i]
An equilateral triangle $ ABC $ is inscribed in a circle; prove that if $ M $ is any point of the circle, then one of the distances $ MA $, $ MB $, $ MC $ is equal to the sum of the other two.
Let $\vartriangle ABC$ be an equilateral triangle with side $1$. Four points are marked $P_1$, $P_2$, $P_3$, $P_4$ on side $AC$ and four points $Q_1$, $Q_2$, $Q_3$, $Q_4$ on side $AB$ (see figure) of such a way to generate $9$ triangles of equal area. Find the length of segment $AP_4$.
[img]https://cdn.artofproblemsolving.com/attachments/5/f/29243932262cb963b376244f4c981b1afe87c6.png[/img]
PS. Easier version of [url=https://artofproblemsolving.com/community/c6h3323141p30741525]2023 Chile NMO L2 P3[/url]
Given equilateral triangle with the side $l$. What is the minimal length $d$ of a brush (segment), that will paint all the triangle, if its ends are moving along the sides of the triangle.
Given three parallel lines, prove that there are three points, one on each line, which are the vertices of an equilateral triangle.
Inside an equilateral triangle there is a point whose distances from the sides of the triangle are $3, 4$ and $5$. Find the area of the triangle.
On the $BE$ side of a regular $ABE$ triangle, a $BCDE$ rhombus is built outside it. The segments $AC$ and $BD$ intersect at point $F$. Prove that $AF <BD$.
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 $m + n$.
Given an equilateral triangle with sidelength $k$ cm. With lines parallel to it's sides, we split it into $k^2$ small equilateral triangles with sidelength $1$ cm. This way, a triangular grid is created. In every small triangle of sidelength $1$ cm, we place exactly one integer from $1$ to $k^2$ (included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths $1$ cm. We shall name as [i]value [/i] of the hexagon, the sum of the numbers that lie on the $6$ small equilateral triangles that the hexagon consists of . Find (in terms of the integer $k>4$) the maximum and the minimum value of the sum of the values of all hexagons .
On the side $BC$ of the equilateral triangle $ABC$, choose any point $D$, and on the line $AD$, take the point $E$ such that $| B A | = | BE |$. Prove that the size of the angle $AEC$ is of does not depend on the choice of point $D$, and find its size.
Consider points $C_1, C_2$ on the side $AB$ of a triangle $ABC$, points $A_1, A_2$ on the side $BC$ and points $B_1 , B_2$ on the side $CA$ such that these points divide the corresponding sides to three equal parts. It is known that all the points $A_1, A_2, B_1, B_2 , C_1$ and $C_2$ are concyclic. Prove that triangle $ABC$ is equilateral.
Points $A_1,B_1,C_1$ are selected on the sides $BC$,$CA$,$AB$ respectively of an equilateral triangle $ABC$ in such a way that the inradii of the triangles $C_1AB_1$, $A_1BC_1$, $B_1CA_1$ and $A_1B_1C_1$ are equal. Prove that $A_1,B_1,C_1$ are the midpoints of the corresponding sides.
Given the isosceles triangle $ABC$, where $AB = AC$. Let $D$ be a point in the segment $BC$ so that $BD = 2DC$. Suppose also that point $P$ lies on the segment $AD$ such that: $\angle BAC = \angle BP D$. Prove that $\angle BAC = 2\angle DP C$.
$ABC$ is a triangle and $D$ is an internal point such that $\angle DAB=\angle DBC =\angle DCA$. $O_a$ is the circumcenter of $DBC$. $O_b$ is the circumcenter of $DAC$. $O_c$ is the circumcenter of $DAB$. Show that if the area of $ABC$ and $O_aO_bO_c$ are equal then $ABC$ is equilateral.
A segment $AB$ is given. We erect the equilateral triangles $ABC$ and $ADB$ above and below $AB$. We denote the midpoints of $AC$ and $BC$ by $E$ and $F$ respectively. Prove that the straight lines $DE$ and $DF$ divide the segment $AB$ into three parts of equal length .
In the rectangle $ABCD$, $AB = 2BC$. An equilateral triangle $ABE$ is constructed on the side $AB$ of the rectangle so that its sides $AE$ and $BE$ intersect the segment $CD$. Point $M$ is the midpoint of $BE$. Find the $\angle MCD$.
In an acute-angled triangle, one of the angles is $60^o$. Prove that the line passing through the center of the circumcircle and the intersection point of the medians of the triangle cuts off an equilateral triangle from it.
(A. Zaslavsky)
In the equilateral triangle $ABC$ ($AB = 2$), cevians are drawn that do not intersect at one point. It turned out that the pairwise intersection points of these cevians lie on the inscribed circle of triangle $ABC$. Find the length of the cevian segment from the vertex of the triangle to the nearest point of intersection with the circle.
a) Prove that a convex $n$-gon cannot have more than $3$ interior angles acute.
b) Prove that a convex $n$-gon that has $3$ interior angles equal to $60^0,$ is equilateral.
In the hexagon $ABCDEF$, all angles are equally large. The side lengths satisfy $AB = CD = EF = 3$ and $BC = DE = F A = 2$. The diagonals $AD$ and $CF$ intersect each other in the point $G$. The point $H$ lies on the side $CD$ so that $DH = 1$. Prove that triangle $EGH$ is equilateral.
An equilateral triangle $ABC$ is given. On the line through $B$ parallel to $AC$ there is a point $D$, such that $D$ and $C$ are on the same side of the line $AB$. The perpendicular bisector of $CD$ intersects the line $AB$ in $E$. Prove that triangle $CDE$ is equilateral.
Let $ABCD$ be rhombus $ABCD$ where the triangles $ABD$ and $BCD$ are equilateral. Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, such that $\angle MAN = 30^o$. Let $X$ be the intersection point of the diagonals $AC$ and $BD$. Prove that $\angle XMN = \angle\ DAM$ and $\angle XNM = \angle BAN$.
Prove that the sides of any equilateral triangle you can either increase everything or decrease everything by the same amount so that you get a right triangle.
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.