All Questions
10 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
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 ...
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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
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
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
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
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
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 $...
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