Questions tagged [quadrilateral]
For questions about general quadrilaterals (including parallelograms, trapezoids, rhombi) and their properties.
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Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers
An exam for high school students had the following problem:
Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a circle be determined by the points $E$, $B$ and ...
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Area of a square inside a square created by connecting point-opposite midpoint
Square $ABCD$ has area $1cm^2$ and sides of $1cm$ each.
$H, F, E, G$ are the midpoints of sides $AD, DC, CB, BA$ respectively.
What will the area of the square formed in the middle be?
I know that ...
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Area of parallelogram = Area of square. Shear transform
Below the parallelogram is obtained from square by stretching the top side while fixing the bottom.
Since area of parallelogram is base times height, both square and parallelogram have the same area.
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Is every parallelogram a rectangle ??
Let's say we have a parallelogram $\text{ABCD}$.
$\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two parallel lines $\text{AB}$ and $\text{CD}$, So, $$ar\triangle \...
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A conjecture involving prime numbers and parallelograms
As already introduced in this post, given the series of prime numbers greater than $9$, let organize them in four rows, according to their last digit ($1,3,7$ or $9$). The column in which they are ...
user559615
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Is there a formula to calculate the area of a trapezoid knowing the length of all its sides?
If all sides: $a, b, c, d$ are known, is there a formula that can calculate the area of a trapezoid?
I know this formula for calculating the area of a trapezoid from its two bases and its height:
$$S=...
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Maximum area of a square in a triangle
I want to calculate the area of the largest square which can be inscribed in a triangle of sides $a, b, c$ . The "square" which I will refer to, from now on, has all its four vertices on the sides of ...
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Japanese theorem for cyclic quadrilaterals Proof Inversion
Two weeks ago our professor taught us the Japanese theorem for cyclic quadrilaterals. It states that the inscribed centres of the four triangles formed by two sides and a diagonal of a cyclic ...
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Trisect a quadrilateral into a $9$-grid; the middle has $1/9$ the area
Trisect sides of a quadrilateral and connect the points to have nine quadrilaterals, as can be seen in the figure. Prove that the middle quadrilateral area is one ninth of the whole area.
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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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Let $ABCD$ be a convex quadrilateral. If the measure of the angles $A=90, C=96, D=78$ and $BC=2*AB$, then the measure of the angle $ABD$ is?
The problem
Let $ABCD$ be a convex quadrilateral. If the measure of the angles $A=90°, C=96°, D=78°$ and $BC=2*AB$, then the measure of the angle $ABD$ is...?
The idea
As you can see I calculated ...
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Angles of triangle $\triangle XYZ$ do not depend on the position of point $P$ (proof needed)
Let $ABCD$ be a fixed convex quadrilateral and $P$ be an arbitrary point. Let $S,T,U,V,K,L$ be the projections of $P$ on $AB,CD,AD,BC,AC,BD$ respectively. Let $X,Y,Z$ be the midpoints of $ST,UV,KL$. ...
user613853
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Can a quadrilateral polygon have 3 obtuse angles?
I was messing around with quadrilaterals trying to draw one that has three obtuse angles. I couldn't create one because with 3 obtuse angles the shape would "open up too much".
I have ...
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Point that divides a quadrilateral into four quadrilaterals of equal area
Consider an irregular quadrilateral $ABCD$. Let $E,F,G,H$ be the midpoints of its edges. It seems that there is a point $K$ such that
$$
S_{AHKE} = S_{EKFB} = S_{KHDG} = S_{KGCF} \left(= \frac{1}{4} ...
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In a parallelogram, does the diagonal bisect the angles that they meet?
I understand the following properties of the parallelogram:
Opposite sides are parallel and equal in length.
Opposite angles are equal.
Adjacent angles add up to 180 degrees therefore adjacent angles ...
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