All Questions
Tagged with polygons euclidean-geometry
35
questions
22
votes
1
answer
2k
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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
1
vote
0
answers
2k
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Exterior angle definition in the case of concave polygons. [duplicate]
Exterior angles are easy to define for convex polygons. They also lie outside the polygon, making it intuitive as to why they are called "exterior". But I'm a bit confused when we talk about exterior ...
1
vote
3
answers
2k
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Area of a cyclic polygon maximum when it is a regular polygon
My question: Let $n$ points $A_1, A_2,\ldots,A_n$ lie on given circle then show that $\operatorname{Area}(A_1A_2\cdots A_n)$ maximum when $A_1A_2\cdots A_n$ is an $n$-regular polygon.
- 1,336
24
votes
5
answers
2k
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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
8
votes
2
answers
364
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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
4
votes
4
answers
3k
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A simple proof that a polygon circumscribing a circle overestimates its perimeter
Looking at the picture below, it's easy to see why the perimeter of a polygon inscribed in a circle is an underestimation of the circle's perimeter. This follows from the triangle inequality: Any side ...
- 24.6k
1
vote
1
answer
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Relationship between the sides of inscribed polygons
In my math textbook there's a demonstration for the calculus of the circumference of a circle that involves regular polygons inscribed in the circle, but I don't get it. The book gives the following ...
- 223
13
votes
2
answers
432
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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
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
8
votes
5
answers
2k
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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
7
votes
0
answers
209
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Inequality conjecture for convex pentagons
Let $X_1, ..., X_5$ be the vertices of a convex pentagon with perimeter $p$ with its centroid at the origin, satisfying $d(X_i, X_j) < \frac{p}{3}$ for $1 \leq i,j \leq 5$, where $d$ is the ...
- 1,460
7
votes
0
answers
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Polygons with coincident area and perimeter centroids
Let $P$ be a simple, planar polygon.
Define $c_a$ as the area centroid of $P$, i.e., the center of gravity of the
closed shape $P$.
Define $c_p$ as the perimeter centroid of $P$,
the center of gravity ...
- 30.4k
7
votes
1
answer
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Proof of $\angle$ sum of polygon.
First, I know this question might have been asked by several times, see here, for an example.
Before someone may want to mark it as dulplicate, I would like to calrify what I want to ask.
Mainly, I ...
- 2,378
6
votes
1
answer
455
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Convex hexagon $ABCDEF$ following equalities $AD=BC+EF, BE=AF+CD, CF=DE+AB$. Prove that $\frac{AB}{DE}=\frac{CD}{AF}=\frac{EF}{BC}$
A convex hexagon $ABCDEF$ is such that the following equalities $AD=BC+EF, BE=AF+CD, CF=DE+AB$ hold. Prove that
$$\frac{AB}{DE}=\frac{CD}{AF}=\frac{EF}{BC}$$
I do not know how to begin to solve ...
- 17.9k