All Questions
Tagged with polygons euclidean-geometry
158
questions
24
votes
5
answers
2k
views
About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals
I've thought about the following question for a month, but I'm facing difficulty.
Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon ...
- 145k
22
votes
1
answer
2k
views
Largest $n$-vertex polyhedron that fits into a unit sphere
In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
user139000
13
votes
4
answers
1k
views
The sum of the squares of the diagonals in a polygon
The first question that got me here:
A regular dodecagon $P_1 P_2 P_3 \dotsb P_{12}$ is inscribed in a circle with radius $1.$ Compute$
{P_1 P_2}^2 + {P_1 P_3}^2 + {P_1 P_4}^2 + \dots + {P_{10} P_{11}}...
- 229
13
votes
2
answers
432
views
Polygons with 2 diagonals of fixed length (part two)
In this question of mine
Polygons with two diagonals of fixed length
I've presented the following particular polygon $P$
and I've asked the following question: is it possible to shorten one or ...
user372511
12
votes
2
answers
675
views
Problem on diagonals in a polygon
Consider the polygon $P$ in the following picture which has sides drawn in black and internal diagonals drawn in red and blue. $P$ has 4 convex angles and 4 concave angles in alternating order as it's ...
user372511
11
votes
3
answers
4k
views
Scaling and rotating a square so that it is inscribed in the original square
I have a square with a side length of 100 cm. I then want to rotate a square clockwise by ten degrees so that it is scaled and contained inside the existing square.
The image below is what I'm ...
- 111
11
votes
1
answer
369
views
Regular polygons constructed inside regular polygons
Let $P$ be a regular $n$-gon, and erect on each edge toward the inside
a regular $k$-gon,
with edge lengths matching. See the example below for $n=12$ and $k=3,\ldots,11$.
Two ...
- 30.4k
10
votes
1
answer
317
views
Sufficient condition to inscribe a polygon inside another one
Let $P$ be any convex polygon in the plane $\mathbb{R}^2$ with vertices $x_1,\dots,x_n$, $n\ge 4$. Let $P'$ be another convex polygon with vertices $x_1',\dots,x_n'$ (same number of vertices of $P$).
...
user372511
9
votes
2
answers
2k
views
If $AB\parallel DC$, $BC\parallel AD$, and $AC\parallel DQ$, find $\Bbb X$ in terms of the areas $\Bbb A$, $\Bbb B$, and $\Bbb C$.
If $AB\parallel DC$, $BC\parallel AD$, and $AC\parallel DQ$, find $\Bbb X$ in terms of the areas $\Bbb A$, $\Bbb B$, and $\Bbb C$.
Please, I wrote a lot of relations, but I just need to prove that
$\...
- 515
9
votes
3
answers
399
views
Show that U,V and H are colinear
We are given a regular icosagon as below:
I wanna prove that the red line exists.
I know that $U$ is the incenter of $\triangle TLB$ ($T,U,G$ are collinear)
I know that $V$ is the incenter of $\...
- 1,940
9
votes
1
answer
207
views
Internal angles in regular 18-gon
This (seemingly simple) problem is driving me nuts.
Find angle $\alpha$ shown in the following regular 18-gon.
It was easy to find the angle between pink diagonals ($60^\circ$). And I was able to ...
- 16k
8
votes
5
answers
2k
views
Fastest method to draw constructible regular polygons
We know from Gauss, that the regular polygons of order $3$, $4$, $5$, $6$, $8$, $10$, $12$, $15$, $16$, $17$, $20$, $24\ldots$ are constructible.
Is there a provably fastest compass and straightedge ...
- 33.1k
8
votes
2
answers
364
views
Convex cyclic hexagon $ABCDEF$. Prove $AC \cdot BD \cdot CE \cdot DF \cdot AE \cdot BF \geq 27 AB \cdot BC \cdot CD \cdot DE \cdot EF \cdot FA$
Convex hexagon $ABCDEF$ inscribed within a circle. Prove that
$$AC \cdot BD \cdot CE \cdot DF \cdot AE \cdot BF \geq 27 AB \cdot BC \cdot CD \cdot DE \cdot EF \cdot FA\,.$$
I was thinking of ...
- 755
8
votes
1
answer
150
views
How to find the $2d$ cross-sections of the $n$-hypercube that are regular $2n$-gons?
I'm currently trying to do something that sounds doable, but on which I've now been stuck for a while.
The idea is that, if you consider an $n$-hypercube, the polygon found by doing a $2d$ cross-...
- 83
8
votes
0
answers
146
views
Which objects can be Minkowski halved?
The Minkowski Sum of two subsets $A,B \subset \mathbb{R}^n$ is
$$A \oplus B = \{a + b | a \in A, b \in B\}$$
For a given $A$, is there some condition that tells me when I can find a $B$ such that $A = ...
- 2,339