Questions tagged [quadrilateral]
For questions about general quadrilaterals (including parallelograms, trapezoids, rhombi) and their properties.
771 questions
4
votes
3
answers
158
views
How to prove that four points are concyclic under these given conditions?
I've stumbled upon the following problem that made me curious for quite a while. Below is one of many possible diagrams of the problem in question.
Given that $ABCD$ is a parallelogram with acute ...
- 429
5
votes
4
answers
362
views
How to construct the following isosceles trapezoid with compass and a straightedge?
Given:
$ABCD$ is an isosceles trapezoid with $BC \parallel AD$.
$MN$ is a segment such that $M\in AB$, $N\in CD$, $BC\parallel MN\parallel AD$.
$AM : MB = DN : NC = 1 : 2$.
$MN = AB = CD$.
$O$ is a ...
- 429
0
votes
2
answers
90
views
Non-concyclicity of the circumcenters of complementary triangles in a quadrilateral
About a year ago, I discovered a property related to quatrains, but I haven't been able to prove it. The property is rather strange, and I don't know where to begin proving it. If anyone can help me, ...
- 4,843
7
votes
2
answers
345
views
What is the length of side AB in quadrilateral ABCD circumscribed about semicircle (O)?
Here's a problem I just came up with :
A semicircle (O) is inscribed in a quadrilateral ABCD , as shown in the figure.
If sides AD , DC , CB measure 17 ; 16 and 14 respectively, what is the length of ...
- 3,518
4
votes
1
answer
364
views
How can this theorem be proven geometrically?
Here's a problem I just came up with:
A chord, passing through the point of intersection of the two diagonals of a cyclic quadrilateral $ABCD$, intersects the circumcircle at $N$ and $P$, as shown in ...
- 3,518
2
votes
1
answer
122
views
A square configuration leading to an isosceles trapezoid and $FB =\frac{na}{(2n+1)(n+1)}$
Let ABCD be a square with side length $a$. Let $M$ be a point on $BC$. The line $AM$ meets the diagonal $BD$ at $K$. From $K$ draw the perpendicular $KL$ to $CD$, with foot $L$. Let $P$ be the ...
2
votes
1
answer
102
views
Prove that diagonals of a cyclic quadrilateral bisects the angle subtend by the midpoint with opposite vertices in harmonic quadrilateral
$ABCD$ is a cyclic quadrilateral. The midpoints of the diagonals $AC$ and $BD$ are respectively $P$ and $Q$. If $BD$ bisects $\angle AQC$, the prove that $AC$ will bisect $\angle BPD$
Source- NMTC ...
- 331
2
votes
1
answer
74
views
Vertices of quadrilaterals as circumcenters
Let $ABCD$ be a quadrilateral. Do there exist points $P,Q,R,S$ such that $A,B,C$ and $D$ are the respective circumcenters of $\triangle QRS$, $\triangle RSP$, $\triangle SPQ$ and $\triangle PQR$?
...
- 331
14
votes
6
answers
3k
views
How to calculate the area of a quadrilateral given the (x,y) coordinates of its vertices
This problem is from the 2017 Gauss Contest (Grade 7).
Four vertices of a quadrilateral are located at (7,6), (−5,1), (−2,−3), and (10,2).
What is the area of the quadrilateral in square units?
I ...
- 1,759
7
votes
2
answers
366
views
Divide a right triangle into three quadrilaterals of equal area
The goal is to find the point $M$ inside a given right triangle $ABC$ such that $\operatorname{Area}(APMN)=\operatorname{Area}(CQMP)=\operatorname{Area}(BQMN)$, where $N$, $P$, and $Q$ are the ...
- 3,518
2
votes
1
answer
166
views
Potentially new formula for the area of a general quadrilateral using all its sides and any one angle
I think this is a potentially new formula for the area of a general quadrilateral. Also do note this post has been repurposed for combining all my 4 posts regarding my formula as they were being ...
- 609
6
votes
0
answers
163
views
Prove concyclic points using Ptolemy's trigonometric theorem
Let $\triangle ABC$ be a scalene triangle with point $D$ inside it. Let $AD$, $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ in $M$, $N$ and $O$. Let the midpoint of the segment that connects the ...
- 613
5
votes
0
answers
119
views
Is this formula correct for the area of any non self intersecting quadrilateral?
$$
\Delta=\frac{1}{2}\left[ab \sin \theta + cd \sqrt{1- \left(\frac {c^2+d^2-a^2-b^2+2ab\cos\theta}{2cd}\right)^2} \right]
$$
This is the formula that I've proven to be able to find the area of any ...
- 609
0
votes
0
answers
166
views
New potential formula for calculating the area of any general quadrilateral
This link shows the derivation of this formula for a concave quadrilateral.
This link shows the derivation of this formula for a convex quadrilateral.
The formula I'm talking about is:
$$
\frac{1}{2}\...
- 609
1
vote
2
answers
193
views
The incenter of a quadrilateral is a rational function of the vertex coordinates
Given only the coordinates of $A,B,C,D$, the incenter can be recovered purely by reflections across diagonals and perpendicular constructions, without ever touching angle bisectors.
Take diagonal $AC$...
- 1