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Questions tagged [quadrilateral]

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

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Isogonal conjugation in quadrilaterals

It's known that there is an isogonal conjugation with respect to triangle. For example, if $P$ is a point and $ABC$ is a triangle then isogonal conjugate of $P$ is defined as point such that $\angle ...
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40 views

Maximizing Area of a quadrilateral inside of a square

The square ABCD has point M located on side AB and point N on side CD. Lines CM and BN intersect at point U. Lines DM and AN intersect at point V. Determine where points M and N should be placed to ...
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184 views

Side length of a quadrilateral incribed on a circle

I've been doing math for 10 years now, yet every so often I get stumped by a "basic" high school question. This is one of those times. Here's the question: Part a is easy; we apply the cosine rule ...
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46 views

$|AB|+|BC|=l$ , find the position of $A,B,C$ maximize area of quadrilateral $OABC$ ? [closed]

The given condition is $|AB|+|BC|=l$ In a 2D coordinate system, point $O(0,0)$ is origin. With the following points: $A(0,a),a\geq0$ $B(b_1,b_2), b_1\geq0,b_2\geq0$ $C(c,0), c\geq0$ and the ...
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44 views

ABCD and AECF are two parallelograms and side EF is parallel to AD . suppose AF and DE met at X and BF AND CE AT Y . prove that XY is parallel to AB

I tried proving it by showing angles exy and eyx equal to edc and ecd respectively but I got no where . Is there any other approach I should consider
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27 views

Rectangle trapezoid

I would be very grateful if you can help me with this problem. I've constructed the median ON, N ∈ BC, and I was able to find that the triangle OCN is isosceles (height and median coincide). Probably ...
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52 views

If the sides of a quadrilateral are $a,b,c,d$, prove that the area cannot exceed $(ac+bd)/2$.

MOP 1997: Let $Q$ be a quadrilateral whose side lengths are $a,b,c,d$ in that order. Show that the area of $Q$ does not exceed $(ac+bd)/2$. My solution: Without loss of generality, let $a$ be the ...
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37 views

Consecutive Vertices of a Quadrilateral

$D, G$ are points on the side $AB$ of $\triangle ABC$. $E$ and $F$ are points on the sides AC and BC respectively such that $DE \parallel BC,$ $EF \parallel AB$ and $FG \parallel CA$. Then $D, E, F, G$...
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58 views

Prove quadrilateral ABCD is a parallelogram

enter image description here In the image, EFGH is a parallelogram, and BE=HC=GD=AF. Can I prove that ABCD is also a parallelogram?
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Find the angle in a quadrilateral

This is the picture, and we are aiming for the angle $x$ It's easy to see that $\angle DGA = \angle CGB = 100°$, $\angle CGD = \angle AGB = 80°$, $\angle CBG = 50°$, but now i'm missing $\angle GBA = ...
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What is the probability if we throw dart towards a large square but it should hit only the inner part of small square $FEHG$ inscribed in it?

Let $ABCD$ be a square shaped board. 4 equal rectangles are drawn into it. The length of the sides of the rectangles are $x$ and $y$, where $\frac{x}{y}$ = $3$. A dart is thrown towards the square ...
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What is the area of the quadrilateral $ADEC$ in $ABC$ right triangle in the following diagram?

In the right angled triangle $ABC$, $\angle A = 90^\circ$, $AB=8$, $AC=6$, $BC = 10$. $D$ is a point on $AB$ in such way that if a perpendicular $DE$ is drawn on $BC$ from $D$ then $BE = 4$. What ...
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A line segment joins the middle points of each diagonal of a quadrilateral with known sides. What is sum of the squares of the diagonals?

$|DF|=|FB|$ $|AE|=|EC|$ What is $|AC|^2 + |BD|^2= \ ?$ I asked one of my classmates for the solution, he said: $$|AC|^2+|BD|^2+4 \cdot3^2 = 7^2 +4^2 +5^2+6^2$$ He didn't tell me how he ...
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4-gon with two opposite sides congruent and two opposite angles congruent: necessarily a parallelogram? [duplicate]

My kid's geometry classmate tried to use the following "theorem" in a proof: If a quadrilateral has a pair of opposite sides congruent and a pair of opposite angles congruent, it's a parallelogram. ...
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Prove a figure is a cyclic quadrilateral

In the figure below, $O$ is the center of the circle. If angle CPB is $90^\circ$, then prove that $AOEF$ is a cyclic quadrilateral.
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Parallel lines & squares

A square $P_1P_2P_3P_4$ has points $X$ on side $P_2P_3$ and $Y$ on side $P_3P_4$ chosen such that angle $XP_1Y$ equals forty-five degrees. The lines $P_1X$ and $P_1Y$ intersect the circumcircle of the ...
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Area of triangles in a quadrilateral

A quadrilateral is divided into 4 triangles with O as the center ( O isn't located in the middle/not by the diagonals). If the area of OAB = 92m², OBC = 84m², and OCD = 108m². What is the area of ODA? ...
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Let $ABCD$ be a cyclic convex quadrilateral such that $AD + BC = AB$. Prove that the bisectors of the angles $ADC$ and $BCD$ meet on the line $AB$.

Let $ABCD$ be a cyclic convex quadrilateral such that $AD + BC = AB$. Prove that the bisectors of the angles ADC and BCD meet on the line $AB$. I tried to find similar triangles since the angles are ...
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A Quadrilateral (The National Mathematical Olympiad in Bulgaria)

Problem: We have the quadrilateral $ABCD$. The middles of $AB$, $BC$, $CD$ and $DA$, are respectively $M$, $N$, $P$ and $Q$. The centroid of $BNP$ is $F$, and the centroid of $NPD$ is $G$. $MG$ ...
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33 views

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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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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62 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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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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39 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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BdMO - 2018 Regional - Geometry 9 [closed]

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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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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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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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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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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38 views

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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229 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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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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37 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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147 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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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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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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262 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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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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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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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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90 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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100 views

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
78 views

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
18 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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94 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.