All Questions
15
questions
3
votes
1
answer
208
views
Find the area of a regular pentagon as a function of its diagonal
For reference:
Calculate the area of a regular pentagon
as a function of its diagonal of length $a$. (Answer:$\frac{a^2}{4}\sqrt\frac{25-5\sqrt5}{2}$)
My progress:
$R$ = radius inscribed circle
$...
- 7,031
1
vote
2
answers
102
views
Are these simple statements about polygons true?
Let $P$ be an $n$-gon with $n \gt 3$. I'm looking for proofs or counterexamples for the following statements:
there exist consecutive vertices $A,B,C$ of $P$ such that $\triangle ABC \cap \partial P =...
- 295
2
votes
1
answer
135
views
Formula for the area of a regular convex pentagon
This question is closely related to my previous question.
Can you provide a proof for the following claim:
In any regular convex pentagon $ABCDE$ construct an arbitrary tangent to the incircle of ...
- 12.9k
-2
votes
2
answers
158
views
Existence of regular $n$-gon whose vertices are arbitrarily close to integer coordinates
I'm self-studying these days about polytopes and I came with this question. I don't know if it's true or not.
Let $\alpha_1$, $\ldots$, $\alpha_n$ angles of convex $n$-gon, $n\not=4$. Prove that for ...
0
votes
1
answer
69
views
Let $A_1A_2....A_n$ be the regular polygon incribed a circle $(O)$ with radius $R$ what is value $k=\frac{\sum_{i<j} A_iA_j^\alpha}{R^\alpha}$?
Let $A_1A_2....A_n$ be the regular polygon incribed a circle $(O)$ with radius $R$, $\alpha$ is real number, let $$k=\frac{\sum_{i<j} A_iA_j^\alpha}{R^\alpha}$$
My question: I am looking for ...
- 247
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
1
vote
4
answers
385
views
Another angle inside a pentagon
I saw this problem recently:
An angle inside a regular pentagon
My question is what would be the geometric constructions to find the $\theta$ angle if we draw $42°$ from the vertices of a regular ...
- 293
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
2
votes
0
answers
193
views
Maximum distance between a point and a convex polygon as a function of its perimeter
Let $P$ be a convex polygon with perimeter $N$, and let $x$ be an arbitrary point. Show that the maximum distance between $x$ and a vertex of $P$ is at least proportional to $N$ (i.e. there exists a ...
5
votes
2
answers
328
views
Simson line in the regular 17-gon
In geometry, given a triangle $ABC$ and a point $P$ on its circumcircle, the three closest points to $P$ on lines $AB$, $AC$, and $BC$ are collinear. The line through these points is the Simson line ...
- 287
2
votes
0
answers
220
views
Reconstructing a polygon from the Midpoints of Its Sides
I was reading through Dijkstra's 'A Collection of Beautiful Proofs' and stumbled upon this elegant piece of work:
11. Reconstructing an odd polygon from the midpoints of its sides.
We shall ...
- 21
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
5
votes
2
answers
810
views
How can $4$ points in the plane be vertices of $3$ different quadrilaterals?
Four points on the plane are vertices of three different quadrilaterals. How can this happen?
The problem is taken from "Kiselev's Geometry - Book I : Planimetry"
At first, I thought it could be ...
- 1,155
1
vote
1
answer
328
views
Find ratio of areas of triangle to pentagon?
ABCDE is a regular pentagon; rays AB and DC intersect at X. Now the area of triangle BCX is 1. What is the area of the pentagon?
I figured out that the area of the pentagon is the square root of 5. (...
- 1,261
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