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.

In a convex cyclic hexagon $ABCDEF$, $BC=EF$ and $CD=AF$. Diagonals $AC$ and $BF$ intersect at point $Q,$ and diagonals $EC$ and $DF$ intersect at point $P.$ Points $R$ and $S$ are marked on the segments $DF$ and $BF$ respectively so that $FR=PD$ and $BQ=FS.$ [b]The segments[/b] $RQ$ and $PS$ intersect at point $T.$ Prove that the line $TC$ bisects the diagonal $DB$.

Let $ABCDEF$ a convex hexagon, and let $A_1,B_1,C_1,D_1,E_1$ y $F_1$ be the midpoints of $AB,BC,CD,$ $DE,EF$ and $FA$, respectively. Construct points $A_2,B_2,C_2,D_2,E_2$ and $F_2$ in the interior of $A_1B_1C_1D_1E_1F_1$ such that both 1. The sides of the dodecagon $A_2A_1B_2B_1C_2C_1D_2D_1E_2E_1F_2F_1$ are all equal and 2. $\angle A_1B_2B_1+\angle C_1D_2D_1+\angle E_1F_2F_1=\angle B_1C_2C_1+\angle D_1E_2E_1+\angle F_1A_2A_1=360^\circ$, where all these angles are less than $180 ^\circ$, Prove that $A_2B_2C_2D_2E_2F_2$ is cyclic. [b]Note:[/b] Dodecagon $A_2A_1B_2B_1C_2C_1D_2D_1E_2E_1F_2F_1$ is shaped like a 6-pointed star, where the points are $A_1,B_1,C_1,D_1,E_1$ y $F_1$.

Given right hexagon. Each side is divided onto $1000$ equal segments. All the points of division are connected with the segments, parallel to sides. Let us paint in turn the triples of unpainted nodes of obtained net, if they are vertices of the unilateral triangle, doesn't matter of what size an orientation. Suppose, we have managed to paint all the vertices except one. Prove that the unpainted node is not a hexagon vertex.

Prove that in an arbitrary convex hexagon there is a diagonal that cuts off from it a triangle whose area does not exceed $\frac16$ of the area of the hexagon. What are the properties of a convex hexagon, each diagonal of which is cut off from it is a triangle whose area is not less than $\frac16$ the area of the hexagon?

A regular hexagon of side $5$ is cut into unit equilateral triangles by lines parallel to the sides of the hexagon. We call the vertices of these triangles knots. If more than half of all knots are marked, show that there exist five marked knots that lie on a circle.

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.

A regular heptagon $ABCDEFG$ is given. The lines $AB$ and $CE$ intersect at $ P$. Find the measure of the angle $\angle PDG$.

Let $ABC$ be a triangle and $O$ be its circumcircle. Let $A', B', C'$ be the midpoints of minor arcs $AB$, $BC$ and $CA$ respectively. Let $I$ be the center of incircle of $ABC$. If $AB = 13$, $BC = 14$ and $AC = 15$, what is the area of the hexagon $AA'BB'CC'$? Suppose $m \angle BAC = \alpha$ , $m \angle CBA = \beta$, and $m \angle ACB = \gamma$. [b]p10.[/b] Let the incircle of $ABC$ be tangent to $AB, BC$, and $AC$ at $J, K, L$, respectively. Compute the angles of triangles $JKL$ and $A'B'C'$ in terms of $\alpha$, $\beta$, and $\gamma$, and conclude that these two triangles are similar. [b]p11.[/b] Show that triangle $AA'C'$ is congruent to triangle $IA'C'$. Show that $AA'BB'CC'$ has twice the area of $A'B'C'$. [b]p12.[/b] Let $r = JL/A'C'$ and the area of triangle $JKL$ be $S$. Using the previous parts, determine the area of hexagon $AA'BB'CC'$ in terms of $ r$ and $S$. [b]p13.[/b] Given that the circumradius of triangle $ABC$ is $65/8$ and that $S = 1344/65$, compute $ r$ and the exact value of the area of hexagon $AA'BB'CC'$. PS. You had better use hide for answers.

Given convex hexagon $ABCDEF$ with $AB \parallel DE$, $BC \parallel EF$, and $CD \parallel FA$ . The distance between the lines $AB$ and $DE$ is equal to the distance between the lines $BC$ and $EF$ and to the distance between the lines $CD$ and $FA$. Prove that the sum $AD+BE+CF$ does not exceed the perimeter of hexagon $ABCDEF$.

Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

Let $\vartriangle ABC$ be an equilateral triangle with side length $M$ such that points $E_1$ and $E_2$ lie on side $AB$, $F_1$ and $F_2$ lie on side $BC$, and $G1$ and $G2$ lie on side $AC$, such that $$m = \overline{AE_1} = \overline{BE_2} = \overline{BF_1} = \overline{CF_2} = \overline{CG_1} = \overline{AG_2}$$ and the area of polygon $E_1E_2F_1F_2G_1G_2$ equals the combined areas of $\vartriangle AE_1G_2$, $\vartriangle BF_1E_2$, and $\vartriangle CG_1F_2$. Find the ratio $\frac{m}{M}$. [img]https://cdn.artofproblemsolving.com/attachments/a/0/88b36c6550c42d913cdddd4486a3dde251327b.png[/img]

Alice and Bob are playing in the forest. They have six sticks of length 1, 2, 3, 4, 5, 6 inches. Somehow, they have managed to arrange these sticks, such that they form the sides of an equiangular hexagon. Compute the sum of all possible values of the area of the hexagon.

Can a regular triangle be placed inside a regular hexagon in such a way that all vertices of the triangle were seen from each vertex of the hexagon? (Point $A$ is seen from $B$, if the segment $AB$ dots not contain internal points of the triangle.)

You have a large number of congruent equilateral triangular tiles on a table and you want to fit $n$ of them together to ma€ke a convex equiangular hexagon (i.e. one whose interior angles are $120^o$) . Obviously, $n$ cannot be any positive integer. The first three feasible $n$ are $6, 10$ and $13$. Determine if $19$ and $20$ are feasible .

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.

A triangle $ABC$ and a point $P$ inside this triangle are given. Define $D, E$ and $F$ as the midpoints of $AP, BP$ and $CP$, respectively. Furthermore, let $R$ be the intersection of $AE$ and $BD, S$ the intersection of $BF$ and $CE$, and $T$ the intersection of $CD$ and $AF$. Prove that the area of hexagon $DRESFT$ is independent of the position of $P$ inside the triangle. [asy] unitsize(1 cm); pair A, B, C, D, E, F, P, R, S, T; A = (0,0); B = (5,0); C = (1.5,4); P = (2,2); D = (A + P)/2; E = (B + P)/2; F = (C + P)/2; R = extension(A,E,B,D); S = extension(B,F,C,E); T = extension(C,D,A,F); draw(A--B--C--cycle); draw(A--P); draw(B--P); draw(C--P); draw(A--F--B); draw(B--D--C); draw(C--E--A); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, N); dot("$D$", D, dir(270)); dot("$E$", E, dir(270)); dot("$F$", F, W); dot("$P$", P, dir(270)); dot("$R$", R, dir(270)); dot("$S$", S, SW); dot("$T$", T, SE); [/asy]

Let $ABCDEF$ be a hexagon inscribed in a circle (with vertices in that order) with $\angle B + \angle C > 180^o$ and $\angle E + \angle F > 180^o$. Let the lines $AB$ and $CD$ intersect at $X$ and the lines $AF$ and $DE$ intersect at $S$. Let $XY$ and $ST$ be the diameters of the circumcircles of $\vartriangle BCX$ and $\vartriangle EFS$ respectively. If $U$ is the intersection point of the lines $BX$ and $ES$ and $V$ is the intersection point of the lines $BY$ and $ET,$ prove that the lines $UV, XY$ and $ST$ are all parallel.

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.

Let $ABCDEF$ be a regular hexagon with sidelength s. The points $P$ and $Q$ are on the diagonals $BD$ and $DF$, respectively, such that $BP = DQ = s$. Prove that the three points $C$, $P$ and $Q$ are on a line. [i](Walther Janous)[/i]

Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE, EF = FA$. Prove that $\frac{BC}{BE} +\frac{DE}{DA} +\frac{FA}{FC} \ge \frac{3}{2}$ . When does equality occur?

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.

In convex hexagon $ABCDEF, AB$ is parallel to $CF, CD$ is parallel to $BE$ and $EF$ is parallel to $AD$. Prove that the areas of triangles $ACE$ and $BDF$ are equal .

The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.

The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB \equal{} BC$, $ CD \equal{} DE$, $ EF \equal{} FA$. Prove that: \[ \frac {BC}{BE} \plus{} \frac {DE}{DA} \plus{} \frac {FA}{FC} \geq \frac {3}{2}. \]