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Questions tagged [quadrilateral]

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

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Construct or prove existence of a certain quadrilateral

I have three questions about a quadrilateral with the following properties: It is convex. It has exactly one pair of congruent opposite sides. It has exactly one pair of congruent opposite angles. It ...
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2answers
86 views

Value of one angle of a quadrilateral, when three sides are quadrilateral are given

In quadrilateral $ABCD$, $AB=BC=CD$. The value of $∠BAC$ is $40º$ and the value of $∠CAD$ is $30º$ what is the value of $∠ADC$?
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4answers
920 views

The equations of two sides of parallelogram are $2x-3y+7=0$ and $4x+y=21$ and one vertex is $(-1, -3)$. Find the other vertices.

I already saw this question here, but the answers are not clear for me. Hope I could get clear solution and answer. I already got the $2^{nd}$ vertex which is $(4, 5)$. How do I get the other two ...
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3answers
992 views

How to prove that the line joinning the midpoints of the diagonals of trapezium is parallel to the parallel sides of the trapezium?

How to prove that the line joining the midpoints of the diagonals of trapezium is parallel to the parallel sides of the trapezium? I tried to prove this by drawing BD and joining the midpoints of AD ...
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9answers
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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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0answers
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Integration along the side in isoparametric quadrilateral mapping

I need to integrate a function (say,$F(x,y)=x^3+y^3$) along one side of the given quadrilateral. For example, the $ x $ and $ y$ coordinate of the quadrilateral are$ (0,0),(2,-1),(3,2),(1,3)$ and the ...
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2answers
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Different possible perimeters

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$.  How many different values of $...
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2answers
98 views

Orthocentre of triangle in cyclic qudrilateral

From vertex B of parallelogram $ABCD$ heights $BK$ and $BH$ are drawn on sides $AD$ and $DC$ respectively. It is known that $KH = a$ and $BD = b$. Find the distance from $B$ to the intersection points ...
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2answers
141 views

Areas in Convex Quadrilateral

$PQRS$ is a convex quadrilateral. The intersection of the two diagonals is $O$. If the areas of triangles $PQS$, $QRP$, and $SPR$ are $1, 2,$ and $3$, respectively. What is the area of triangle $PQO$?
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1answer
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The quadrilateral X has vertices at A = (0,1), B= (2,0), C=(6,1), and D= (6,4). How would you fill up the plane with the shape X as much as possible?

So I have the following image of a shape X: It's asking how to fill the plane with as much shape X as possible. I'm guessing it's asking me how to align it so that there is least amount of free space....
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1answer
68 views

How to know if a quadrilateral is circumscribed

Let's say I have a quadrilateral that is convex and I know the points that are forming it. How can I know if the quadrilateral is circumscribed? Is it true that a quadrilateral is circumscribed if and ...
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1answer
275 views

Interesting tiling with a lot of symmetrical shapes

I have such an interesting observation: if I take a square grid and rotate it over itself by atan(3/4) , it forms a structure which has four axes of reflection symmetry: The resulting structure is ...
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2answers
235 views

Any employment for the Varignon parallelogram?

The midpoints of the sides of an arbitrary quadrilateral form a parallelogram, which is called the Varignon parallelogram of the quad. While answering a question about Quadrilateral Interpolation it ...
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1answer
607 views

Geometry Question on Quadrilaterals [Triangle and Square]

Q) ABCD is a square. Prove that Triangle ABP is equilateral My attempt:as PDC is isosceles, PD = PC, AD = BC and ∠ADP = ∠BCP therefore, Triangles ADP and BPC are congruent so BP=AP.. now i don't get ...
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3answers
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Quadrilateral Interpolation

The simplest finite element shape in two dimensions is a triangle. In a finite element context, any geometrical shape is endowed with an interpolation, which is linear for triangles (most of the time),...
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1answer
154 views

If all four angles inside a quadrilateral are defined, how to find the angle of intersection of diagonals of the quadrilateral?

Say, I have a quadrilateral where all four internal angles are defined, say ∠a, ∠b, ∠c, ∠d . This should be enough to determine the shape of the quadrilateral even though its size can be arbitrary. ...
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1answer
163 views

Triangles, Quadrilaterals

In triangle ABC, D and E are the point on AC and AB respectively. BD and CE intersect at F. If the areas of triangles EBF, BFC, FDC are 10, 20, 16 respectively. Then area of quadrilateral AEFD is?
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1answer
206 views

How to find inner angles and side of the rhombus given height and diagonal AC

I have to find inner angles $\alpha$, $\beta$, $\delta$ and $\gamma$, other diagonal $\left|BD\right|$ and length of a side $a$ into a rhombus where is given the height $h = 2\sqrt{3} cm$ and the ...
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1answer
503 views

point equidistant from all sides of quadrilateral

If $ABCD$ is a quadrilateral in which $AB+CD=BC+AD$, Prove that the bisectors of the angles of the quadrilateral meet in a point which is equidistant from the sides of the quadrilateral. Firstly,I ...
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2answers
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Radius of a cyclic quadrilateral given diagonals

I noticed something curious about intersecting chords in a circle. Suppose two chords have lengths $p$ and $q$ and intersect at right angles at point $O$. The intersection $O$ divides the two chords ...
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1answer
115 views

Centre of ellipse with inscribed trapezoid?

Is there a way to find out the centre coordinates of an ellipse given the corner points of a trapezoid inscribed within it? The height and the lengths of AB and CD are known. Here is a diagram: In ...
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1answer
878 views

prove that the quadrilaterals are congruent

If 2 quadrilaterals ABCD and PQRS have angles A,B,C,D equal to angles P, Q, R, S respectively and AB=PQ and CD=RS and is AD is not parallel to BC prove that the quadrilaterals are congruent. I was ...
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1answer
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Show that ABQC is a cyclic quadrilateral & $\triangle BPD $ is isosceles

Hi can anyone help me out here , Because I struggling in iii & iv What I have done so far $i.\angle BAC=2a$ $ii. \angle BAQ= \angle QAC$ $\therefore \angle BCQ = \angle QAC$ (alternate segment ...
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1answer
855 views

Check for collinearity between four points that supposedly build a quadrilateral

Let's say we get four 2D points (let's name these $p1$, $p2$, $p3$ and $p4$) from somewhere and we want to build a quadrilateral (doesn't matter if convex or not). What is the optimal check for ...
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5answers
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Prove that the perimeter of any quadrilateral is greater than twice the length of any of its diagonal

I am stuck with the following problem that says : Prove that the perimeter of any quadrilateral is greater than twice the length of any of its diagonal. My try: ........ For any quadrilateral ...
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2answers
609 views

Largest incircle inside a quadrilateral radius calculation

How can I calculate the radius of the biggest possible inscribed circle that is inside any quadrilateral? Every quadrilateral can have an incircle that is adjacent to at least 3 sides right? I want to ...
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1answer
358 views

Circumcentre of triangles in a quadrilateral

Suppose the diagonals of a quadrilateral ABCD meet at a point O; prove that the circumcentres of the 4 triangles OAB,OBC,OCD, and ODA form a parallelogram.
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2answers
2k views

In convex quadrilateral $ABCD,$ $\angle A \cong \angle C,$ $AB=CD=180,$ and $AD \ne BC.$ The perimeter of $ABCD$ is $640.$ Find $\cos A.$

In convex quadrilateral $ABCD,$ $\angle A \cong \angle C,$ $AB=CD=180,$ and $AD \ne BC.$ The perimeter of $ABCD$ is $640.$ Find $\cos A.$ I have no idea on where to start on this problem
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2answers
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Minimizing Perimeter of a quadrilateral

Let us say that we are given a quadrilateral where the diagonals are congruent and fixed at a certain length, and the angle between the two diagonals are fixed. How would you prove that the minimum ...
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1answer
196 views

Using Slopes and Distances to Determine Quadrilaterals

Ok, so I recently came across this problem: "Use slopes and distances to determine what kind of quadrilateral PQRS is created by each set of coordinate points. $P(0,0), Q(0,2), R(5,5), S(2,0)$ $P(1,...
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0answers
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Find the area of a quadrilateral [closed]

How to find the maximum possible area of a convex quadrilateral given that its perimeter is 5m and diagonals are 2m ? I have tried the Heron's formula for quadrilaterals, but it's too complicated.
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1answer
135 views

Orthodiagonal quadrilateral

Is it always possible to construct an orthodiagonal quadrilateral such that the diagonals and perimeter are fixed? More specifically, given 2 fixed diagonals, how is the perimeter of the quadrilateral ...
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2answers
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Proof With Cyclic Quadrilateral and Circle

Let $ABCD$ be a cyclic quadrilateral. Let $P$ be the intersection of $\overline{AD}$ and $\overline{BC}$, and let $Q$ be the intersection of $\overline{AB}$ and $\overline{CD}$. Prove that the angle ...
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1answer
382 views

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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1answer
247 views

Let ABCD be a cyclic quadrilateral…

Let ABCD be a cyclic quadrilateral. Let r and s be the lines obtained by reflecting AB through the angle bisectors of $\angle CAD$ and $\angle CBD$, respectively. Let P be the intersection of r and s ...
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1answer
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Show that three circles are coaxal

Let $A_1, A_2, A_3, A_4$ are collinear, $B_1, B_2, B_3, B_4$ are collinear. Such that $A_1, A_2, B_2, B_1$ lie on circle $(O_1)$, and $A_3, A_4, B_4, B_3$ lie on circle $(O_2)$. Let $MNPQ$ be the ...
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3answers
183 views

Find a the coordinate of a missing point (R) in quadrilateral [closed]

$PQRS$ is a quadrilateral with vertices $P(1,3)$ $Q(5,5)$ $ R(x,1) $ $S(4,-3). $ Question: Find the coordinates of $R$ if the area of quadrilateral $PQRS$ is $42$ square unit . How to find $x$ in ...
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3answers
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Where do you see cyclic quadrilaterals in real life?

I've just been studying cyclic quads in geometry at school and I'm thinking see seems pretty interesting, but where would I actually find these in the real world? They seem pretty useless to me...
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0answers
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Finding position of 3D point constrained by “parent” point constrained to circle and rotation

I have the following 3D geometry question from a camera positioning problem: (Related to this question which I got an answer to, but I was not able to extrapolate to 3D) Point $P_1$ (parent) can ...
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1answer
36 views

Finding position of 2D point constrained by “parent” point constrained to circle and rotation

I have the following 2D geometry question from a camera positioning problem: Point $P1$ (parent) can only be on a circle about the origin with given radius $R$. Point $P2$ (child)'s position is ...
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0answers
257 views

Is one diagonal of a convex quadrilateral always longer than at least 3 sides?

I am solving a problem which requires a proof of the statement in the title. So far, I was considering the following cases: 4 angles are at least 90°: rectangle, the statement holds. 3 angles are at ...
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1answer
50 views

Find the measure of $DME$

In the parallelogram $ABCD$ $A$ is the small angle and $BC=2AB$. We draw the height $CE$ to $AB$.We draw a line from $C$ and $E$ to $M$. if $M$ is the midpoint of $AD$ find the measure of $DME$ in ...
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0answers
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Convex quadrilateral with interior point

Let $ABCD$ be a convex quadrilateral and $X$ is an interior point. Also let $AX\cap BD=\{E\}$, $BX\cap AC=\{F\}$, $CX\cap BD=\{G\}$ and $DX\cap AC=\{H\}$. Prove that: $$AF\cdot BG\cdot CH\cdot DE=...
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1answer
140 views

A triangle in a square

The following quadrilateral is a square also there are some known angles.prove that The segments of the inner triangle are equal. My Attempt:If we name the inner point $O$ then two triangles $AOD$ ...
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6answers
934 views

Given distances (shortest paths) between four cities, how to show that they cannot be in the same plane?

In the example below we are given distances between four cities. The author of the book says that these distances "suffice to prove that the world is not flat". Do I understand this correctly that ...
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3answers
406 views

proof that diagonals of a quadrilateral are perpendicular if $AB^2+CD^2=BC^2+AD^2$

proof that diagonals of a quadrilateral are perpendicular if $AB^2+CD^2=BC^2+AD^2$. My Attempt:we know that if diagonals of a quadrilateral are perpendicular then we have $AB^2+CD^2=BC^2+AD^2$.But ...
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2answers
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In a parallelogram, do the diagonal bisect the angles that they meet?

I understand the following properties of a 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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1answer
982 views

Show that if AB+CD=AD+BC then the quadrilateral $ABCD$ is tangential

Consider a convex quadrilateral $ABCD$ . Show that the quadrilateral $ABCD$ is tangential if and only:$AB+CD=AD+BC$
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1answer
465 views

How do you find 2 variables in a quadrilateral, only knowing the other two?

If you know the angles of 2 sides of an irregular quadrilateral, how do you find out the other 2? Many Thanks, Alex
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2answers
46 views

Question about angles of rhoumbus

Problem: Consider a rhombus (Diamond) such that each of its side is the geometric mean of its diameters. I mean if length of each side is X and the diameters a and b; then $X^2$ = a.b Find the ...