For questions about general quadrilaterals (including parallelograms, trapezoids, rhombi) and their properties.

372 questions
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Calculating area of quadrilateral when distance of vertices from an arbitrary point is known

Given a convex quadrilateral $ABCD$ circumscribed about a circle of diameter $1$. Inside $ABCD$ there is a point $M$ such that $MA^2 + MB^2 +MC^2 + MD^2 =2$. Find the area of the quadrilateral. My ...
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Mapping from a square $\left[-\frac{1}{2},\frac{1}{2}\right]\times\left[-\frac{1}{2},\frac{1}{2}\right]$ with local coordinate system $\,(\xi,\eta)\,$ to an arbitrary quadrilateral with global ...
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Quadrilateral Finite Elements must be convex and not self-intersecting. But why?

Main reference @ Mathematics Stack Exchange: Quadrilateral Interpolation Quoted from this question: Why a quadrilateral with bilinear interpolation? Little else is possible with polynomial terms ...
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Finding the area of a parallelogram given the length of its diagonals and their intersection angle

The diagonals of a parallelogram have lengths of $15.6 \text{cm}$ and $17.2 \text{cm}$. They intersect at an angle of $120$. Find the area of the parallelogram. The part I find most confusing is the ...
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Finding unknown angles of a rhombus given side length and area

Given that the area of a rhombus is $40 \text{cm}^2$ and that each side has a length of $15 \text{cm}$, find the angles of the rhombus. It's from a 8th-grade school math textbook.
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Area of square as a function of $\hat{a}$?

Suppose $ABCD$ is square ,and $AM=DN=QB=PC$ so $$A'B'C'D'$$ is a square too. Can someone help me to find area of $\bf{smaller -square}$(or $\color{red} {\Box A'B'C'D'}$) as a function of ...
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Construct triangle ABC, denote I as the incenter, A' as the mid point of the arc BC of the circumcircle. Show that A'B = A'C = A'I

I know angles BAC and BCA are equal since the arcs BA and AC are equal however I do not know where to go on from there. Hints or answers involving cyclic quadrilaterals would be appreciated. Thank you....
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How to inscribe an ellipse into an isosceles trapezoid?

My main question is: how can I inscribe an ellipse into an isosceles trapezoid? I want to create an ellipse, whick is tanget to all four sides of the trapeziod (i.e. shares exactly one point with each)...
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Parallelogram and Congruence

Let point M be outside the parallelogram ABCD such that $\angle MAB = \angle MCB$. Prove that $\angle AMD = \angle CMB$. I am trying to prove $\triangle MDE \sim \triangle MBC$ but I am having ...
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Finding the fourth point of a perfect square (without knowing order of points)

When i was writing my programming project the other day i ran into an interesting problem that i couldn't solve, i spent a while trying solutions with absolute values but none of that worked. Here's ...
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Iterating a geometric construction on quadrilaterals

Let $ABCD$ be any quadrilateral, and let $r, s, t, u$ be its four angle bisectors. If the $r,s,t,u$ are not concurrent (as in the image below on the left), then they intersect to form the vertices of ...
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Geometry Proof Concerning Equality of Lengths on a Quadrilateral

Quadrilateral $WXYZ$ has right angles at $\angle W$ and $\angle Y$ and an acute angle at $\angle X$. Altitudes are dropped from $X$ and $Z$ to diagonal $\overline{WY}$, meeting $\overline{WY}$ at $O$ ...
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Proof of inequality of area for quadrilateral

If $a,b,c,d$ are sides of a quadrilateral, and $S$ is its area, then prove that $$\dfrac{(a+c)^2+bd}{4} \geq S$$.
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How do I know if this quadrilateral is cyclic?

There are some pairs of angles that are equal to each other, but none of the values are known.
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Please don't judge by pointing to my other questions here, but all I got is that the lower angle is 45 deg. Interior angles sum to 180(n-2), I think, sooo... $$360 -45 = 180(n-2)-45 = 2x+y$$ where I ...
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Distinct convex quadrilaterals formed by 12 points out of which 5 are collinear

There are 12 points in a plane of which 5 are collinear. The number of distinct convex quadrilaterals which can be formed with vertices at these points is:___ I know how to solve this question ...
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To find area of quadrilateral $ABCD$ using the given co-ordinates. [closed]

I am unable to get the answer to this question. The question is to find the area of a quadrilateral having its vertices as coordinates in order: $A(3,-2)$; $B(4,0)$; $C(6,-3)$ and $D(5,-5)$. I ...
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Angle bisector in a trapezoid - surface area ratio

In trapezoid ABCD (AB || CD) the angle bisector of angle ABC is perpendicular to segment AB and intersects it in point P. Point P divides the side AD in ratio 2:1. Find the ratio of the surface areas ...
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Given area of quadrilateral find missing coordinates

Can someone please explain how to proceed with this question, or maybe give me hints as to how to do it? I am not familiar with concave polygons at all, so any help with that as well would be greatly ...
Let two circles intersect at $X$ and $Y$, and let a common tangent touch the circles at $P$ and $Q$. A third circle is drawn so that it is tangent to the two circles at $A$ and $B$. Let the line $XY$ ...