Found problems: 200
The hexagon $ABCDEF$ is convex and opposite sides are parallel. Show that the triangles $ACE$ and $BDF$ have equal area
Angles are equal in a hexagon, three main diagonals are equal and the other six diagonals are also equal. Is it true that it has equal sides?
A positive integer $n$ is [i]Danish[/i] if a regular hexagon can be partitioned into $n$ congruent polygons. Prove that there are infinitely many positive integers $n$ such that both $n$ and $2^n+n$ are Danish.
Suppose that in a convex hexagon, each of the three lines connecting the midpoints of two opposite sides divides the hexagon into two parts of equal area. Prove that these three lines intersect in a point.
a) Points $A_1, A_2, A_3, A_4, A_5, A_6$ divide a circle of radius $1$ into six equal arcs. Ray $\ell_1$ from $A_1$ connects $A_1$ with $A_2$, ray $\ell_2$ from $A_2$ connects $A_2$ with $A_3$, and so on, ray $\ell_6$ from $A_6$ connects $A_6$ with $A_1$. From a point $B_1$ on $\ell_1$ the perpendicular is drawn on $\ell_6$, from the foot of this perpendicular another perpendicular is drawn on $\ell_5$, and so on. Let the foot of the $6$-th perpendicular coincide with $B_1$. Find the length of segment $A_1B_1$.
b) Find points $B_1, B_2,... , B_n$ on the extensions of sides $A_1A_2, A_2A_3,... , A_nA_1$ of a regular $n$-gon $A_1A_2...A_n$ such that $B_1B_2 \perp A_1A_2$, $B_2B_3 \perp A_2A_3$,$ . . . $, $B_nB_1 \perp A_nA_1$.
Let $ABCDEF$ be a hexagon with sides $AB,CD,EF$ being equal to $m$ units, sides $BC,DE, FA$ being equal to $n$ units. The diagonals $AD,BE,CF$ have lengths $x, y$, and $z$ units. Prove the inequality $$\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx} \ge \frac{3}{(m+ n)^2}$$
Nguyễn Văn Quý
A plane passing through the center of a cube intersects the cube in a cyclic hexagon. Show that this hexagon is regular.
Let $ABCDEF $ be a regular hexagon. On the line $AF $ mark the point $X$so that $ \angle DCX = 45^o$ . Find the value of the angle $FXE$.
(Vyacheslav Yasinsky)
The length of each side and each non-principal diagonal of a convex hexagon does not exceed $1$. Prove that this hexagon contains a principal diagonal whose length does not exceed $\frac{2}{\sqrt3}$.
Given convex hexagon $ABCDEF$, inscribed in the circle. Prove that
$AC*BD*DE*CE*EA*FB \geq 27 AB * BC * CD * DE * EF * FA$
Let $ ABCDEF$ be a convex hexagon with $ AB \equal{} BC \equal{} CD$ and $ DE \equal{} EF \equal{} FA$, such that $ \angle BCD \equal{} \angle EFA \equal{} \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB \equal{} \angle DHE \equal{} \frac {2\pi}{3}$. Prove that $ AG \plus{} GB \plus{} GH \plus{} DH \plus{} HE \geq CF$.
Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.
About the convex hexagon $ABCDEF$ it is known that $AB = BC =CD = DE = EF = FA$ and $AD = BE = CF$. Prove that the diagonals $AD$, $BE$, $CF$ intersect at one point.
The hexagon $ABLCDK$ is inscribed and the line $LK$ intersects the segments $AD, BC, AC$ and $BD$ in points $M, N, P$ and $Q$, respectively. Prove that $NL \cdot KP \cdot MQ = KM \cdot PN \cdot LQ$.
Let $F$ be the boundary and $M,N$ be any interior points of a triangle $ABC$. Consider the function $f_{M,N}: F \to R$ defined by $f_{M,N}(P) = MP^2 +NP^2$ and let $\eta_{M,N}$ be the number of points $P$ for which $f{M,N}$ attains its minimum.
(a) Prove that $1 \le \eta_{M,N} \le 3$.
(b) If $M$ is fixed, find the locus of $N$ for which $\eta_{M,N} > 1$.
(c) Prove that the locus of $M$ for which there are points $N$ such that $\eta_{M,N} = 3$ is the interior of a tangent hexagon.
Given $7$ points inside or on a regular hexagon, show that three of them form a triangle with area $\le 1/6$ the area of the hexagon.
Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$
Let the regular hexagon $ABCDEF$ be inscribed in a circle with center $O$, $N$ be such a point Let $E-N-C$, $M$ a point such that $A- M-C$ and $R$ a point on the circumference, such that $D-N- R$. If $\angle EFR = 90^o$, $\frac{AM}{AC}=\frac{CN}{EC}$ and $AC=\sqrt3$, calculate $AM$.
Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.
On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle at the point of impact.
A regular hexagon with its vertices on the circle is drawn on a circular billiard table.
A (point-shaped) ball is placed somewhere on the circumference of the hexagon, but not on one of its edges.
Describe a periodical track of this ball with exactly four points at the rails.
With how many different directions of impact can the ball be brought onto such a track?
Let $ABCDEF$ be a convex hexagon satisfying $AC = DF, CE = FB$ and $EA = BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.
Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom?
[img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]
A diagonal of a convex hexagon is called [i]long[/i] if it decomposes the hexagon into two quadrangles.
Each pair of [i]long[/i] diagonals decomposes the hexagon into two triangles and two quadrangles.
Given is a hexagon with the property, that for each decomposition by two [i]long[/i] diagonals the resulting triangles are both isosceles with the side of the hexagon as base.
Show that the hexagon has a circumcircle.
In the convex $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively.
a) Show the inequalities $\frac{1}{2} \le \frac{s}{u+v}\le 1$
b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons.
A football is made by sewing together some black and white leather
patches. The black patches are regular pentagons of the same size. The white
patches are regular hexagons of the same size. Each pentagon is bordered by 5
hexagons. Each hexagons is bordered by $3$ pentagons and $3$ hexagons. We need $12$ pentagons to make one football.
How many hexagons are needed to make one football?
A parallelogram $ABCD$ is split by the diagonal $BD$ into two equal triangles. A regular hexagon is inscribed into the triangle $ABD$ so that two of its consecutive sides lie on $AB$ and $AD$ and one of its vertices lies on $BD$. Another regular hexagon is inscribed into the triangle $CBD{}$ so that two of its consecutive vertices lie on $CB$ and $CD$ and one of its sides lies on $BD$. Which of the hexagons is bigger?
[i]Konstantin Knop[/i]