Questions tagged [quadrilateral]
For questions about general quadrilaterals (including parallelograms, trapezoids, rhombi) and their properties.
771 questions
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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 ...
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Is this result already a known theorem in geometry?
I have been playing around with geometry and I found that:
Let two perpendicular lines intersect at a point that is inside a circle. Then the area of the quadrilateral formed by the vertices made by ...
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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 ...
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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, ...
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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 ...
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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 ...
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Find the angle of ACD
I want to find $\angle ACD$. Given quadrilateral $ABCD$ as picture above. Let $DC=DB=AB$. Given $\angle ADB=78^\circ, \angle DAC=48^\circ, \angle CAB=30^\circ, \angle ABD=24^\circ$.
I just can only ...
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$a^2+b^2=c^2+d^2$ and $a^2+d^2−ad=b^2+c^2+bc$. Find the nearest integer value of the expression $\frac{ab+cd}{ad+bc}$
If $a,b,c,d$ are positive reals such that
$a^2+b^2=c^2+d^2$ and $a^2+d^2−ad=b^2+c^2+bc$,
find the nearest integer value of the expression
$\frac{ab+cd}{ad+bc}$.
I have seen a solution. It goes like ...
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How to calculate projection matrix for quadrilateral transform?
I have a square and its 4 corner coordinates $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$. And I have a quadrilateral with corner coordinates $(x_1',y_1'),(x_2',y_2'),(x_3',y_3'),(x_4',y_4')$ where $(x_i,...
CommunityBot
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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 ...
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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 ...
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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 ...
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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 $\triangle PQR$?
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Intuition behind reflecting $D$ in IGO $2023$ geometry problem
Problem: Let $ABCD$ be a convex quadrilateral. Let $E$ be the intersection of its diagonals.
Suppose that $CD = BC = BE$. Prove that $AD + DC ≥ AB$.
My approach: I tried to make a triangle which has a ...
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Find an angle in the given quadrilateral
In the following problem, I want to find the angle marked as $x$. It seems so simple and yet I am out of ideas. It is very easy to get all angles except two of them: angle ADB and angle CBD.
Is ...
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