Questions tagged [quadrilateral]

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

7
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2answers
1k views

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 ...
0
votes
1answer
711 views

Jacobian determinant for bi-linear Quadrilaterals

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 ...
0
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2answers
188 views

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 ...
0
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2answers
4k views

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 ...
0
votes
2answers
2k views

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.
2
votes
2answers
53 views

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 ...
0
votes
2answers
294 views

A relation between sides of quadrilateral

Suppose $ABCD$ a quadrilateral (in Euclidean geometry), with $$ \overline{AD} \leq \overline{AC},\\ \overline{BC} \leq \overline{BD}. $$ Then show that $$ |\overline{AD} - \overline{BC}| \leq \...
0
votes
1answer
306 views

Relationship of aspect ratio to the homography matrices between a rectangle and an arbitrary quadrilateral

I've been reading everything I can on the perspective mapping between a 2D rectangle and the projection onto the plane in 3D space of a rectangle. I've learned that any such quadrilateral resulting ...
0
votes
1answer
150 views

Can every arbitrary set of four points in 2D space map to at least one right-angled rectangle perspective-projected from 3D space onto the plane?

I have an intuition that given a rectangle of arbitrary width and height rotated arbitrarily in 3D space and perspective-projected onto the 2D plane, that not all arbitrary sets of resulting 2D points ...
1
vote
3answers
8k views

If ABCD is a quadrilateral in which AB || CD and AD=BC, prove that $\angle$A=$\angle$ B.

Q. Let ABCD be a quadrilateral in which AB || CD and AD=BC. Prove that $\angle$A=$\angle$ B. My attempt: Connecting BD and AC and trying to prove $ \Delta ADC \cong \Delta BCD $. In $ \Delta ADC \...
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2answers
93 views

In a rhombus, $ABCD$ with diagonals intersecting at $o$, Prove that $AB^2 + BC^2 + CD^2 + DA^2 = 4(OA^2 + OB^2)$ [closed]

How to solve this following problem: In a rhombus, $ABCD$ with diagonals intersecting at $o$, Prove that $AB^2 + BC^2 + CD^2 + DA^2 = 4(OA^2 + OB^2)$.
2
votes
3answers
87 views

A synthetic proof that a figure with 4 right angles is planar

Assume that we have a skew quadrilateral ABCD, such that the four angles are right angles (a rectangle). How can I prove synthetically that ABCD are coplanar ?
3
votes
2answers
176 views

Prove that $\gamma = 99^{\circ}$ in this quadrilateral.

$ABCD$ is a quadrilateral. $\measuredangle{BAD}=86^{\circ}$ and $\measuredangle{CDA}=68^{\circ}$, $|AB|=|CD|$, $E$ and $F$ are midpoints of their segments. $\measuredangle{DEF}=\gamma$. Prove that $\...
0
votes
1answer
212 views

A circle is inscribed in trapezoid $ABCD(BC \parallel AD)$. The circle is tangent to the sides of $AB$ and $CD$ at $K$ and $L$, respectively.

A circle is inscribed in trapezoid $ABCD(BC \parallel AD)$. The circle is tangent to the sides of $AB$ and $CD$ at $K$ and $L$, respectively, and to bases $AD$ and $BC$ at $M$ and $N$, respectively. ...
4
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6answers
2k views

How to find missing angles in a quadrilateral

I have a quadrilateral ABCD, with diagonals AC and BD. Given are four angles: ∠DAC = 20°, ∠CAB = 60°, ∠ABD = 50°, and ∠DBC = 30°. Those are the red angles in the above image. I need to fill in all ...
3
votes
3answers
228 views

angle chasing in quadrilaterals

Let $ABCD$ be a convex quadrilateral with $\measuredangle{ABD} = 18^{\circ}$, $\measuredangle{ACB} = 54^{\circ}$, $\measuredangle{ACD} = 36^{\circ}$ and $\measuredangle{ADB} = 27^{\circ}$...
0
votes
1answer
80 views

Angles and Quadrilaterals Problem 70 [closed]

In a quadrilateral FGHI, FG = GI, ∠GFI = 70, ∠GHI = 55 and FI is parallel to GH. What is the size of ∠GFH? I thought it would be 50. Is that correct?
9
votes
7answers
933 views

What is a simple (or not) way of finding the lengths of the diagonals of this rhombus?

I recently taught a group of geometry students about properties of rhombuses, and gave them a set of homework exercises created by a previous instructor which included the following problem. If a ...
-3
votes
2answers
261 views

Irregular quadrilateral [closed]

I have an irregular quadrilateral. I know the length of three sides (a, b and c) and the length of the two diagonals (e and f). All angles are unknown How do I calculate the length of the 4th side (d)?...
1
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0answers
44 views

Making a Triangle from $3$ sides of a Quadrilateral

This problem has stumped me for a little while: Given a quadrilateral with integer side lengths, what is the probability that three of its sides can be removed and reassembled to form a triangle? ...
0
votes
2answers
354 views

Sufficient condition for quadrilateral to be cyclic

It's straightforward to show that if quadrilateral $ABCD$ has its vertices lying on a common circle, then its opposite angles complement one another. That is $\angle DAB + \angle DCB =\angle ADC + \...
2
votes
1answer
81 views

In a quadrilateral $ABCD, $ it is given that $AB$ is parallel to $CD$…

In a quadrilateral $ABCD, $ it is given that $AB$ is parallel to $CD$ and the diagonals are perpendicular to each other. Show that (i)$AD\cdot BC\ge AB\cdot CD$ (ii)$AD+BC\ge AB+CD$ I have ...
4
votes
1answer
200 views

$ABCD$ is a quadrilateral and $P,Q$ are midpoints of $CD, AB.$…

$ABCD$ is a quadrilateral and $P,Q$ are midpoints of $CD, AB.$ $AP$ and $DQ$ meet at $X, BP$ and $CQ$ meet at $Y.$ Prove that $$|ADX|+|BCY|=|PXQY|$$ (here $|N|$ means area of the shape $N$) I have ...
2
votes
1answer
73 views

Advanced Level Geometry prob.

$ABCD$ is parallelogram. $$m(AEB) = 79°$$ $$m(ABD) = x$$ How do I find $x$? I have tried to find it and got $102°$ but I'm wrong. Thanks for helping.
2
votes
1answer
87 views

Help me find a side of a quadrilateral given one side and distance between the centers of two circles inscribed in the same quadrilateral

For the quadrilateral ABCD it's given that AD=2, $\angle$ABD=$\angle$ACD=90$^\circ$. The distance between the centers of the inscribed circles in $_\triangle$ABD and $_\triangle$ACD is $\sqrt{2}$. ...
2
votes
1answer
294 views

What shape is this quadrilateral?

It is a quadrilateral but I do not know which one it is:
0
votes
1answer
791 views

What is measure of a quadrilateral with an equalateral triangle within the shape?

http://i.imgur.com/URCItTG.jpg BEC is an equilateral triangle and angle ABC is 130, what is the measure of angle ADC. The answer I was given is 50 degrees, however I am curious as to steps required ...
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votes
2answers
55 views

A diagonal in a quadrilateral [closed]

A convex quadrilateral $ABCD$ with sides $AB=8$ cm, $BC=16$ cm, $CD=4$ cm and $AD=6$ cm is given. Find the diagonal $BD$ if the length is an integer.
1
vote
1answer
547 views

Finding the parallel sides of a trapezoid given all side lengths and height from base

Suppose that we are given side lengths $a, b, c, d$ of a trapezoid. We know that two of them are parallel, and all values are different. Moreover, we are given the height $h$ from the base (distance ...
-2
votes
1answer
471 views

Why do we have circles for ellipses, squares for rectangles but nothing for triangles?

migrate to math ed se if need be please. i'm lazy =) A circle is an ellipse with equal foci. A square is a rectangle with equal sides. Why is there no special name for equilateral triangle? Context ...
0
votes
1answer
627 views

Ways to Prove the Converse of Ptolemy's Theorem

This is the proof that I have for Ptolemy's Theorem: $\triangle ABC \sim \triangle ADM$ by AA so $\displaystyle\frac{AB}{AD} = \frac{AC}{AM} = \frac{BC}{DM}$ $\triangle ABD \sim \triangle ACM$ by ...
1
vote
1answer
49 views

Using Ptolemy's Theorem to find length

In $\triangle ABC$ we have $AB=7, AC=8, BC=9$. Point $D$ is the midpoint of the arc $BC$ of the circumcircle of $\triangle ABC$. Compute $\displaystyle\frac{AD}{BD}$, $BD$, and $CD$. This is what I ...
2
votes
2answers
92 views

Cyclic Quadrilateral and Ptolemy to find the length of a segment

$ABCD$ is a cyclic quadrilateral with $AB=11$ and $CD= 19$. Points $P$ and $Q$ are on $AB$ and $CD$ respectively such that $AP=6,\, BP=5\, DQ=7$ and $CQ=12,\, PQ=27$. Extend $PQ$ till it meets the ...
1
vote
2answers
1k views

Length of two sides in a quadrilateral with given angles

I'm stuck finding the length of two sides in a quadrilateral for which I know all angles and the length of two sides. All red objects are know ($a,b,\alpha,\beta,\gamma $ and $\delta$). I need to ...
3
votes
1answer
326 views

Elementary Geometry: Prove that ABCD is a tangential quadrilateral!

The task is as follows: ABCD is a convex quadrilateral with points E, F, G and H on its sides AB, BC, CD and DA so that they divide the respective side at the ratio of the adjacent sides, so $$\...
0
votes
2answers
154 views

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....
1
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2answers
205 views

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)...
1
vote
2answers
161 views

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 ...
2
votes
3answers
117 views

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 ...
3
votes
0answers
126 views

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 ...
1
vote
3answers
142 views

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 $O$ ...
2
votes
0answers
102 views

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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2answers
632 views

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.
0
votes
1answer
475 views

Find largest angle in quadrilateral

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 ...
0
votes
1answer
536 views

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 ...
1
vote
5answers
2k views

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 ...
1
vote
1answer
210 views

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 ...
0
votes
3answers
401 views

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 ...
9
votes
1answer
162 views

The smallest parallelogram that contains a convex quadrilateral

I try to find the smallest parallelogram in terms of area that contains a convex quadrilateral(A,B,C,D). I am pretty sure it must be constructed from two neighboring sides of the quadrilateral. But ...
1
vote
1answer
114 views

Three circles - two intersecting and the third touching

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 $XY$ ...