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

372 questions
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Let $AL$ and $BK$ be the angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects line $AL$ at $M$. The ...
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The bisector of angle $BAD$ in parallelogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. Prove that the center of the circle passing through the points $C$, $K$, ...
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Finding shaded triangle areas in a parallelogram

There is the following parallelogram involving two shaded triangles. If I found rightly, angles of $AMD$, $BMN$ and $CDM$ are $45$. But I can’t go further.
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Is this result already a known theorem in geometry?

I have been playing around with geometry and I found that: Let two perpendicular lines intersect at a point that is inside a circle. Then the area of the quadrilateral formed by the vertices made by ...
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What is the $\angle EBF$?

$ABCD$ is a square. If $\angle EFB= \angle BFC$ what is $\angle EBF?$ I can only think of the Z-rule and say that $\angle BFC = \angle FBA$. After that I can't progress any further. I think the ...
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Why are parallelograms defined as quadrilaterals? What term would encompass polygons with greater than two parallel pairs?

It seems the definition of a parallelogram is locked to quadrilaterals for some reason. Is there a reason for this? Why couldn't a parallelogram (given the way the word seems rather than as a ...
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Find the maximum area of the quadrilateral inscribed on $y=2x-x^2$, where $y\geq 0$ and explain your answer. I can just estimate the shape but I don't know how to prove it precisely. Help me with a ...
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Maximize area of a quadrilateral given three sides

What is the maximum possible area that a quadrilateral can have, if the lengths of three of its sides are given as 3, 4 and 5, while the fourth side can have arbitrary length? (Thinking of it as three ...
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A problem in Euclidean geometry (similarity of two triangles)

Consider a trapezoid $ABCD,$ with major basis $AB,$ circumscribed to a circle of radius $R.$ Let $F$ be the intersection of the lines $AD$ and $BC.$ Choose the point $E$ on the line $CD,$ on the side ...
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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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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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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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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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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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$D, G$ 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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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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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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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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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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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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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 ...