Found problems: 11
Let $ABCDEFG$ be a regular heptagon. We'll call the sides $AB$, $BC$, $CD$, $DE$, $EF$, $FG$ and $GA$ opposite to the vertices $E$, $F$, $G$, $A$, $B$, $C$ and $D$ respectively. If $M$ is a point inside the heptagon, we'll say that the line through $M$ and a vertex of the heptagon intersects a side of it (without the vertices) at a $\textit{perfect}$ point, if this side is opposite the vertex. Prove that for every choice of $M$, the number of $\textit{perfect}$ points is always odd.
Let $ABCDEFG$ be a regular heptagon. If $AC = m$ and $AD = n$, prove that $AB =\frac{mn}{m+n}$.
The set $M$ consists of $n$ points on the plane and satisfies the conditions:
$\bullet$ there are $7$ points in the set $M$, which are vertices of a convex heptagon,
$\bullet$ for arbitrary five points with $M$, which are vertices of a convex pentagon, there is a point that also belongs to $M$ and lies inside this pentagon.
Find the smallest possible value that $n$ can take .
Determine the greatest possible value of the constant $c$ that satisfies the following condition: for any convex heptagon the sum of the lengthes of all it’s diagonals is greater than $cP$, where $P$ is the perimeter of the heptagon.
(I. Zhuk)
How many axes of symmetry can a heptagon have?
The $7$-gon $A_1A_2A_3A_4A_5A_6A_7$ is inscribed in a circle. Prove that if the centre of the circle is inside the $7$-gon , than $$\angle A_1+ \angle A_2 + \angle A_3 < 450^o$$
A regular heptagon $A_1A_2... A_7$ is inscribed in circle $C$. Point $P$ is taken on the shorter arc $A_7A_1$.
Prove that $PA_1+PA_3+PA_5+PA_7 = PA_2+PA_4+PA_6$.
Is there a heptagon and a point $P$ inside it such that any vertex of the heptagon has its distance from $P$ equal to the length of the side opposite the vertex?
[i]A side and a vertex are said to be opposite if the side is the fourth from the vertex page (in any direction).[/i]
Two regular polygons, a $7$-gon and a $17$-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal. (3)
Given a regular heptagon $A_1...A_7$. Prove that $$\frac{1}{|A_1A_5|} + \frac{1}{|A_1A_3| }= \frac{1}{|A_1A_7|}$$.
A regular heptagon $A_1A_2A_3A_4A_5A_6A_7$ is given. Straight $A_2A_3$ and $A_5A_6$ intersect at point $X$, and straight lines $A_3A_5$ and $A_1A_6$ intersect at point $Y$. Prove that lines $A_1A_2$ and $XY$ are parallel.