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Questions tagged [quadrilateral]

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

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How to calculate the co-ordinates of a quadrilateral knowing the side lengths

Hi I've come across a mathematical problem, which I can't seem to solve with my limited geometry and trigonometry knowledge. I need to draw a quadrilateral (may not be rectangle always). To draw this ...
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2answers
463 views

Where does this property involving quadrilaterals come from?

$ABCD$ is a square. $|AF|=6$, $|FK|=2$, and $DE \parallel AB$. What is $|EK|=?$ My geometry book has a property for this: $$|AF|^2=|FK|\cdot|FE|$$ Can you show me where does this property come from ...
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1answer
55 views

Show that if $AR$ intersects midperpendicular of $MN$ at $X$, then $X\in(I)$

Triangle ABC has an incircle $(I)$ which contacts $BC,CA,AB$ at $D,E,F$. Let $BP,CQ$ be bisectors of $\angle ABC,\angle ACB$ ($P \in AC,Q\in AB$). Line $AI$ intersects circle $(I)$ at $J$ (point $J$ ...
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Find area of quadrilateral in triangle. [closed]

What is the area of $HIJK$ quadrilateral, if the area of $ABC$ triangle is $70$, $BE=ED=DA$, and $BF=FG= GC$?
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2answers
55 views

Construct a quadrilateral, not a parallelogram, in which pair of opposite angles and a pair of opposite sides are equal. [duplicate]

I want to construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal. I tried drawing one, but I am not able to. Please help. (...
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1answer
38 views

Does the equality of the sum of opposite sides in a quadrilateral necessarily imply the existence of an inscribed circle?

I had come across a question in which we had to show that a given quadrilateral, if subjected under a condition, had an incircle. So, will it be sufficient to show that $a+b=c+d$, if $a,b,c,d$ are the ...
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2answers
41 views

BdMO - 2018 Regional - Geometry 9 [on hold]

In $ABCD$ tetragon $E $ and $F $ are mid points of $AB$ and $AD$ respectively. $CF$ intersects $BD$ at point $G$. If $\angle FGD = \angle AEF$ and the area of $ABCD$ is $24$ , what is the area of $...
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A line segment with a length of 24 makes a 90-degree angle with one of the legs of an isosceles trapezoid. What is the area of this Trapezoid?

Given that $ABCD$ is an isosceles trapezoid and that $|EB|=24$, $|EC|=26$, and m(EBC)=$90^o$. Find $A(ABCD)= ?$ From the pythagorean theorem, I can find that $|BC|=|AD|=10$. Then, I can find the area ...
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Volume of an Irregular Octahedron from edge lengths?

Does anyone know how to calculate the volume of an irregular octahedron from the lengths of the edges? The octahedron has triangular faces, but the only information are the edge lengths. ...
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1answer
50 views

Calculate surface normal and area for a non-planar quadrilateral

Given the four coordinates of the vertices, what is the best possible approximation to calculate surface area and outward normal for a quad? I currently join the midpoints of the sides, thus ...
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Given a trapezoid with base $AD$ larger than side $CD$. The bisector of $\angle D$ meets $AB$ at $K$. Prove $AK > KB$

We have a trapezoid $ABCD$ with base $AD$ larger than side $CD$. The bisector of $\angle D$ intersects side $AB$ at point $K$. Prove that $AK>KB$. All that I have tried was to make such drawing in ...
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1answer
38 views

Proving a quadrilateral is cyclic

I am given that, for $JG$ an exterior angle bisector of $\angle CGF$ parallel to the angle bisector of $\angle FHE$, prove that $CDEF$ is a cyclic quadrilateral. I can prove that if the quadrilateral ...
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1answer
25 views

Area of a right quadrilateral

Quadrilateral $ABCD$ has right angles only at vertices $A$ and $D$. The numbers show the areas of two of the triangles. What is the area of $ABCD$? The rectangle $DABC'$ will have an area of $30$. I ...
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2answers
59 views

Proof that 4 points lie on a circle and that center of this circle lies on the circumcircle of $\triangle ABC$

Given is acute triangle $ABC$. Let $D$ be foot of altitude from vertex $A$. Let $D_1$ be a point so that line of symmetry between $D_1$ and $D$ is line $AB$. Let $D_2$ be a point so that line of ...
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1answer
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Finding coordinate in a quadrilateral

Im trying to do a simulation using Matlab to solve some fluid problem. For this problem I have the following shape: enter image description here For each black point I know the (x,y) coordinates. I ...
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2answers
141 views

Arbelos and its angle bisector

I have recently been reading about a very interesting geometry problem and have tried to solve it. I'm now in a point, in which I don't know how to move forward and would appreciate if someone could ...
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2answers
27 views

Given two circles, prove the following

See picture I was able to identify the cyclic quadrilateral that exists, but not sure how to prove this using that information
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1answer
31 views

Is it possible to invert the Japanese Theorem for cyclic quadrilaterals?

I am very curious, if it is possible to invert the Japanese Theorem. So, if the middlepoints of the inner circle create a rectangle, is the quadrilateral on the outside of the four circles always a ...
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2answers
86 views

Rhombus in a cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral whose opposite sides are not parallel. The lines $AB$ and $CD$ intersect at point $P$. The lines $AD$ and $BC$ intersect in point $Q$. The bisector of the angle $\...
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In the quadrilateral ABCD , the points M and N are the centers of opposite sides AB and DC. [duplicate]

In the quadrilateral ABCD , the points M and N are the centers of opposite sides AB and DC, let MD and AN intersect each other at the point Q and let MC and BN intersect each other at the point R. ...
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1answer
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I have to show that $P, R, Q, S $ are on a circle.

Let $ABCD $ a paralelogram and $H $ the hortocenter of $\triangle ABC $. Let $PQ $ , $RS $ trough $H $ s.t. $PQ|| AB $ and $RS||BC $ and $P\in [DA], R\in [AB], Q\in [BC] , S\in [CD] $. I have to ...
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2answers
36 views

Prove that here $BE×DF$ is fixed

We have a parallelogram namely $ABCD$. Then we draw a line from the vertex $A$ to: 1- Cross a point (namely $E$)on the side $BC$. 2- Cross a point (namely $F$) along the side DC. Now we must prove ...
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4answers
255 views

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 $ST,UV,KL$. ...
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1answer
61 views

Prove that $NK$ is tangent to the circumcircle of $\Delta KEF$ .

Consider a circle $O$ with radius $R$. $ABCD$ is cyclic quadrilateral, the intersection of $AC$ and $BD$ is $K$. $P$ and $Q$ are respectively the midpoints of $KD$ and $KC$. The intersection of $AP$ ...
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1answer
47 views

Diagonals of convex $\square ABCD$ meet at $O$. Show that $|\triangle AOB| = |\triangle COD|$ if and only if $BC \parallel AD$

Let $O$ be the intersection of the diagonals in a convex quadrilateral $\square ABCD$, Show that $|\triangle AOB| = |\triangle COD|$ (that is, the areas are equal) if and only if $BC$ is parallel to $...
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1answer
48 views

Do these areas have any special meaning or name?

I was just toying around with circles and squares. Do the red areas in the picture have any special meaning somehow?
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74 views

The concave quadrilateral and the slopes of its sides

Suppose a plane quadrilateral ABCD without sides parallel to y-axis, and let $m_1, m_2, m_3, m_4$ be the slopes of the equations of sides AB, BC, CD, DA (the cartesian axes being orthogonal or ...
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1answer
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A concave quadrilateral cannot be circumscribed by an ellipse nor by a parabola.

It's intuitive that a concave quadrilateral cannot be circumscribed by an ellipse nor by a parabola. I can prove it by analytic geometry: Without loss of generality, let $L_1\equiv m_1x -y +r_1=0$, $...
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2answers
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The cyclic quadrilateral and the slopes of its sides

Suppose a plane quadrilateral ABCD (convex, concave or crossed) no side of which is parallel to y-axis, and let $m_1, m_2, m_3, m_4$ be the slopes of the equations of sides AB, BC, CD, DA. Having ...
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1answer
16 views

Maximum value of largest side

If we have a quadrilateral with integral and distinct sides and second largest side is 10 then the value of largest side is? And, we can solve it by saying that other two sides will be 9 and 8 ...
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2answers
64 views

Distance between two circles [closed]

See image. In situation like this. Is AE and EC always equal. if no then in which cases are the two lines equal.
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27 views

Unable to relate the given sources/set up equation with the given info to solve quadrilateral problem

$ABCD$ is a square with area 625, $CDEF$ is a rhombus with an area of 500, area of the shaded region is $55x$. Find $x$ wherein $x$ is a single digit non-zero number.
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2answers
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Slopes of perpendicular diagonals of a quadrilateral and its sides

Let ABCD be a quadrilateral, let E be the intersection of lines AB and CD, and let F be the intersection of lines BC and AD. If the lines AC and BD of this quadrilateral are perpendicular and the ...
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2answers
70 views

Solve irregular quadrilateral given 2 angles and 3 sides

Given a follwing irregular quadrilateral: We know 3 sides (a,2a,c) and 2 angles (one right angle and alpha). Need to find side b and two other angles.
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1answer
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Proving two sides of a convex quadrilateral are equal for given condition.

The length of each side of a convex quadrilateral ABCD is a positive integer. If the sum of the lengths of any three sides is divisible by the length of the remaining side then prove that some two ...
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0answers
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Help in proving the inequality related to convex quadrilateral

I have the following problem with me: "Inside the convex quadrilateral ABCD, in which AB = CD, the point P is chosen in such a way that sum of the angles PBA and PCD is 180. Prove that PB + PC < ...
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2answers
62 views

Quadrilaterals and parallel lines: An olympiad question

Let $ABCD$ be a convex quadrilateral $\measuredangle ADC = \measuredangle BCD > 90$. Let $E$ be the point in which line $AC$ intersects the line parallel to $AD $ through $B$ and Let $F$ be the ...
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0answers
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Equations of the diagonals of a quadrilateral (convex, concave or crossed) from the equations of its sides, without calculating its vertices

Let ABCD be a plane quadrilateral (convex, concave or crossed), let E be the intersection of lines AB and CD, let F be the intersection of lines BC and DA, let G be the intersection of diagonals AC ...
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1answer
37 views

If angles $A$, $B$, $C$ of convex quadrilateral $\square ABCD$ are equal, then $D$ lies on the Euler line of $\triangle ABC$

In a convex quadrilateral $ABCD$ angles at $A,B,C$ are equal. Prove that vertex $D$ lies on the Euler line of triangle $ABC$. My try: We can use complex numbers. Set circumcirle of triangle $ABC$ as ...
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0answers
81 views

Beautiful cyclic quadrilateral property involving isogonal conjugates.

Let $ABCD$ be a cyclic quadrialetral. Diagonals $AC$ and $BD$ intersect at point $S$. Denote midpoint of $AC$ with $M$. Choose points $P\in MD$ and $Q\in MB$ so that $PQ\parallel BD$ (in other words ...
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1answer
39 views

A convex quadrilateral with propotion

Let P be the intersection of the convex quadrilateral ABCD. Let X,Y,Z be points on AB,BC,CD respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that XY is tangent to the ...
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1answer
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Prove the identity in inner product space

Could you help me to prove the identity in inner product space below? Since I learned cauchy-schwarz-inequality and quadrilateral identity, but I can't prove it. $\ (w_1 - w_2)^T(w_3 - w_4) = \frac{1}{...
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1answer
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Given two trapezoids of equal area, show that the length of FB is (a^2 + b^2) / 2

In the accompanying figure, i am told that the area of trapezoid BCGF equals the area of trapezoid FGED. With BC = a, and DE = b, i am to prove that the length of FG is sqrt((a^2 + b^2) / 2). Here is ...
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2answers
65 views

cyclic quadrilateral inside an isosceles right triangle

Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\overline{BC}$. Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, ...
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3answers
39 views

Can you find $2$ sides of a quadrilateral with $2$ sides and all $4$ angles?

Not too great with math so sorry if this isnt even possible. Basically im trying to find the 2 sides that are black. I know the angles and sides that are marked in red. The 2 non right angles are not ...
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1answer
312 views

The hunting of “missing primes”

First, I would like to introduce a peculiar way to display the prime numbers (greater than $9$) by means of the ten they belong to ($x$-axis), and their ending digit ($y$-axis). Here's an example of ...
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3answers
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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 ...
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1answer
42 views

Tangential quadrilaterals description

Take a quadrilateral $ABCD$ and consider a line parallel to each of its side such that: 1- the distance between the parallel line and the side is some fixed amount $x$ and 2- the parallel line are "...
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1answer
49 views

inequality proving for intersection

AB and CD are line segments, $AB=CD=1$, intersecting in point O, $\enspace$ $AB\cap CD =O$, $\angle AOC=60^{\circ}$. Prove that $AC+BD\geq1$. $\enspace$ What I tried: $AO+BO=1$, $\enspace$ $CO+DO=1$ ...
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1answer
78 views

Finding an Angle using Cyclic Quadrilateral and Circle Theorems

To find angle BGE from the diagram above, a proof was proposed: "Angle BCF equals 100 degrees (external to triangle ACB); ADEC and DBFC are cyclic quadrilaterals -> angle ADE equals 100 degrees; ...