All Questions
Tagged with polygons euclidean-geometry
158
questions
0
votes
0
answers
25
views
Rational quantities associated with a bicentric heptagon
For odd reasons of my own, I would like to create a math puzzle involving a bicentric heptagon (i.e., a 7-sided polygon which has both an inscribed and a circumscribed circle). The puzzle is to be ...
- 547
0
votes
0
answers
26
views
Polygon Boundary in 3D Space
A simple polygon (interior included) is a manifold with boundary being homeomorphic to a closed disk. Its (manifold) boundary is a non-intersecting closed polygonal chain. Viewed as a subset of the ...
2
votes
1
answer
290
views
Constructing bicentric pentagon
I'm trying to construct a bicentric pentagon in geogebra. I read on Wikipedia that a Pentagon is bicentric if and only if it satisfies this formula $$r(R-x)=(R+x)\left(\sqrt{(R-r)^2-x^2}+\sqrt{2R(R-r-...
- 41.4k
0
votes
1
answer
57
views
Finding the Areas of Polygons from Side Lengths
I am aware of the formula for the area of a regular polygon:
$A=([Side Count] \times [Side Length] \times [Apothem Length])/2$
However, I could not find an equation for the area of a non-regular ...
- 4,368
1
vote
0
answers
2k
views
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 ...
- 30.4k
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}}...
- 24.5k
3
votes
1
answer
106
views
Olympiad Trapezoid Problem about Lengths
I'd like some help with the following Olympiad Problem about a trapezoid:
There is a trapezoid $ABCD$ with parallel sides $BC$ and $AD$ such that $AB=1$, $BC=1$, $CD=1$ and $DA=2$. Let $M$ be the ...
- 1,975
1
vote
1
answer
69
views
Getting a point in the interior of a polygon without relying on winding order?
I am given an arbitrary set of points embedded in 3D.
The points are guaranteed to be ordered such that their order yields a simple closed polygon, but there is no information about whether they wind ...
- 97.7k
3
votes
2
answers
355
views
Collinearity in bicentric pentagon
Can you provide a proof for the following claim:
Claim. The circumcenter, the incenter, and the excenter of the pentagon formed by diagonals in a bicentric pentagon are collinear.
GeoGebra applet ...
- 8,836
5
votes
1
answer
258
views
What is a gyrational square in this context?
This is a diagram of quadrilaterals and their duals. Within this diagram, there is a square (shown with 8 lines of symmetry) and right below that there is a "gyrational square" shown with no ...
- 5,692
7
votes
2
answers
241
views
How to enumerate unique lattice polygons for a given area using Pick's Theorem?
Pick's Theorem
Suppose that a polygon has integer coordinates for all of its vertices. Let $i$ be the number of integer points interior to the polygon, and let $b$ be the number of integer points on ...
- 3,019
4
votes
2
answers
397
views
If all the sides of an n sided polygon are equal. Is it always a regular polygon?
For an $n$-sided polygon, if all the sides are equal, is it a regular polygon?
If yes then why is it defined to have equal angles?
If not so, how to prove that all angles are equal?
edit: I meant it ...
- 16.3k
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 ...
0
votes
3
answers
101
views
Construction of a pentagon in which an angle bisector at a certain vertex is also a perpendicular bisector of a side opposite to that vertex
Construct a pentagon $ABCDE$ if lengths of its sides are $a,b,c,d$ and $e$ ($|AB|=a,|BC|=b,...$), and the bisector of the angle at vertex $D$ is also the perpendicular bisector of $AB$.
I know that $...
- 78.1k
-1
votes
3
answers
87
views
length of side of a regular $n$-gon is less than length of any diagonal
In any regular polygon with $n\ge 4$ sides, why is any side length strictly length to any diagonal length? (A diagonal is defined as the line segment joining non-adjacent vertices)
This is intutitvely ...
- 2,753