All Questions
23 questions
6
votes
4
answers
584
views
Necessary and sufficient condition for quadrilateral to be cyclic
Can you provide a proof for the following proposition:
Proposition. Given any quadrilateral $ABCD$. Let $P,Q,R,S$ be nine-point centers of triangles $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ ...
- 2,661
6
votes
1
answer
224
views
Necessary and sufficient condition for tangential polygon to be cyclic
Can you prove or disprove the following claim?
Claim. Let $A_1,A_2, \ldots ,A_n$ be the vertices of an $n$-sided tangential polygon and let $B_1,B_2, \ldots ,B_n$ be the contact points of the ...
- 2,661
11
votes
2
answers
863
views
Strange formula for area of a convex polygon
Consider a convex $n-$gon in $\mathbb{R}^2$ with sides contained in the lines $y=k_ix+b_i, 1\leq i\leq n.$ Then its area equals to
$$
S=\frac{1}{2}\sum_{i=1}^{n} \frac{(b_{i+1}-b_i)^2}{k_{i+1}-k_i}.
$$...
- 4,316
3
votes
1
answer
158
views
The product of the lengths of two line segments that belong to Newton line [closed]
I am looking for the proof of the following claim:
Consider a family of bicentric quadrilaterals with the same inradius length and the same distance between incenter and circumcenter. Denote by $P$ ...
- 2,661
1
vote
1
answer
111
views
Collinearity in tangential pentagon [closed]
I am looking for a proof of the following claim:
Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of ...
- 2,661
1
vote
1
answer
317
views
A generalization of Harcourt's theorem
This question is closely related to my previous question.
Can you prove the claim given below? The following claim is a conjectured generalization of Harcourt's theorem.
Claim. Let $A_1,A_2 \ldots ...
- 2,661
4
votes
1
answer
320
views
Collinearity in bicentric polygons
Can you provide a proofs for the following two claims?
Claim 1. The circumcenter, the incenter, and the intersection of the principal diagonals in a bicentric even-sided polygon are collinear.
Claim ...
- 2,661
6
votes
1
answer
880
views
Relation of some Euclidean geometry theorems and more conjecture generalizations
In this topic I want to share relation of the Pythagorean theorem, the Stewart theorem and the British Flag theorem, the Apollonius' theorem, the Ptolemy's theorem and the Feuerbach-Luchterhand. Since ...
- 1,044
3
votes
1
answer
805
views
Brother of Japanese theorem for cyclic quadrilaterals
I am looking for a proof of a like result as follows and Higher-dimensional generalizations?
Let $A, B, C, D$ be four point with lengths of $AB, BC, CD, DA$ are $a, b, c, d$ respectively. Let $F \in ...
- 1,776
3
votes
1
answer
123
views
Collinearity of three significant points of bicentric pentagon
Can you provide a proof for the following claim?
Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from ...
- 2,661
1
vote
1
answer
320
views
A formula for the area of bicentric quadrilateral
Can you provide a proof for the claim given below? The following claim is inspired by Harcourt's theorem and can be seen as its generalization to quadrilaterals.
Claim. Given bicentric quadrilateral $...
- 2,661
3
votes
1
answer
103
views
Equal sums of line segments
I would like to see a proof of the following
Claim. Let $A_1,A_2,A_3,A_4,A_5$ be vertices of bicentric pentagon. Let $B_1$ be the intersection point of $A_1A_3$ and $A_2A_5$, $B_2$ the intersection ...
- 2,661
1
vote
1
answer
84
views
Equal products of triangle areas
Can you prove the following claim:
Claim. Given hexagon circumscribed about an ellipse. Let $A_1,A_2,A_3,A_4,A_5,A_6$ be the vertices of the hexagon and let $B$ be the intersection point of its ...
- 2,661
3
votes
1
answer
303
views
How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle
How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle?
See also:
Malfatti circles
- 1,776
2
votes
0
answers
83
views
Principal diagonals of octagon meet in a single point
Can you provide a proof for the following claim:
Claim. Given octagon circumscribed about an ellipse. If the vertices of the octagon lie on another ellipse then its principal diagonals meet in a ...
- 2,661
8
votes
2
answers
339
views
Angle subtended by the shortest segment that bisects the area of a convex polygon
Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...
- 113
1
vote
0
answers
84
views
How can construct the equilateral $A''B''C''$ such that area of $A''B''C''$ is biggest
Let $ABC$ be arbitrary triangle in a plane. Let $A'B'C'$ and $A''B''C''$ be two equilateral triangles such that $A \in B'C'$, $B \in C'A'$, $C \in A'B'$ and $A \in B''C''$, $B \in C''A''$, $C \in A''B'...
- 1,776
4
votes
1
answer
283
views
Reordering vertices of a polygon
Let $Q,Q'$ be two planar polygons with the same number $n>3$ of vertices. There is a correspondence between vertices of $Q$ and $Q'$: to any vertex $z$ of $Q$ corresponds a unique vertex $z'$ of $Q'...
user101163
13
votes
1
answer
921
views
Limiting shape for Brillouin zones
Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle?
You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...
- 12.3k
7
votes
1
answer
5k
views
Shrink polygon to a specific area by offsetting
I have a 2D polygon that I want to shrink by a specific offset (A) to match a certain area ratio (R) of the original polygon. Is there a formula or algorithm for such a problem? I am interested in a ...
- 171
2
votes
5
answers
6k
views
Quadrilateral from 4 random points
Given 4 random points in 2D, how do I compute the area of the quadrilateral formed by the points?
I'm aware of formulae giving the area when I know the sides a,b,c,d and the diagonals p & q.
But ...
- 123
6
votes
1
answer
767
views
Using mirrors to make a non-convex polygon visible from a fixed interior point
Take a point $A$ inside a non-convex polygon $P$. Is it always possible to place a finite set of mirrors given by straight segments (not necessarily along the boundary of $P$, any position inside $P$ ...
- 17.9k
6
votes
1
answer
715
views
Elementary problem about triangles inside a convex polygon
Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy ...
- 1,318