Questions tagged [quadrilateral]
For questions about general quadrilaterals (including parallelograms, trapezoids, rhombi) and their properties.
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Prove that a convex quadrilateral $ABCD$ with certain properties is a cyclic quadrilateral [closed]
$ABCD$ is a convex quadrilateral such that $\angle ABD = \angle DBC$, $AD=CD$ and $AB \neq BC$. Prove that $ABCD$ is cyclic.
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Proving that the cyclic quadrilateral $ABCD$ is a parallelogram, and thus it is a rectangle.
Let us say that we have a circle and a cyclic quadrilateral $ABCD$ where the segment $BD$ passes through the center of the circle, thus being the diameter. Also, $\angle A$ and $\angle C$ are both ...
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Geometric construction of an ellipse enscribed within a in irregular quadrilateral
I am trying to construct the faces of a cube in 3 point perspective, with ellipses enscribed in the same way as shown in this post. I can only use a straight edge and compass, but I can construct an ...
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Diagonals in an inscribed quadrilateral can simultaneously be angle bisectors of triangles involving their midpoints
This is follow-up of an interesting question asked some days ago on this site, that the asker has erased 24 hours later.
The initial question was (as reflected in my title) in terms of an inscribed ...
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Find the area of the quadrilateral $ABDC$
If $A=(2,1), B(8,1), C(4,3), D(6,6)$ then find the area of the quadrilateral $ABDC$.
My Attempt:
Area of quadrilateral= area of triangle ABD + area of triangle ADC.
Area of triangle ABD= $\frac12|2(1-...
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Ellipses inscribed in parallelograms $y=\pm 1$, $y=m(x\pm1)$.
Lately I saw many questions about ellipses inscribed in quadrilaterals in Math Stack Exchange.
By substituting the equations of lines in the general equation
$$ax^2+bxy+cy^2=1$$
of an (maybe) ellipse ...
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Find all inscribed ellipses in a given convex quadrilateral passing through a given internal point
Given a convex quadrilateral, and a point inside it, I want to find all ellipses that are inscribed in the quadrilateral and passing through the given point.
My attempt: is outlined in my solution ...
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Value of angles of a quadrilateral [closed]
$ABCD$ is a rectangle, $\overline{AC}$ and $\overline{BD}$ are its two diagonals, $O$ their intersection point, and $\angle COD=68^{\circ}$. What is the value of $\angle (BAO-OBC)$?
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ellipse inscribed in a quadrilateral
An ellipse with foci $E,F$ is tangent to four sides of quadrilateral $ABCD$, then
$$\tag{eq1} AB⋅EC⋅DF=FA⋅BE⋅CD$$
The order of points in the equation is interesting: $ABECDF$ is a cyclic shift of $...
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sum of square root is constant for all isogonal conjugates wrt a quadrilateral
I am interested in a question by @guangzhou-2015 with no solutions. Equivalent form:
An ellipse with foci $E,F$ is tangent to four sides of quadrilateral $ABCD$, then
$$\sqrt{EA \times EC \times FA \...
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Is Basic Proportionality theorem applicable to trapezium? [closed]
I was studying vectors and came up with this question
Show that the line joining the midpoints of two non parallel sides of a trapezium is parallel to the parallel sides and is equal to half of their ...
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How do you build a quadrilateral, knowing it's sides, their order, and that this quadrilateral has an inscribed circle?
I want to build a quadrilateral with sides, for example, $2:3:5:4$, knowing that it also has an inscribed circle (which all 4 sides are tangent to). How can you do it for a general case, using only a ...
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How to prove $EN=\dfrac{AI}{2}?$ and $KN=\dfrac{CJ}{2}$? and $\dfrac{Area(ABD)}{2}+\dfrac{Area(BCD)}{2}=\dfrac{Area(ABCD)}{2}$?
Given
\begin{aligned}
\operatorname{Area}(E F G H) & =E K \cdot F G \\
& =(E N+K N) \cdot \frac{1}{2} B D \\
& =\frac{1}{2} B D \cdot E N+\frac{1}{2} B D \cdot K N \\
& =\frac{1}{4} ...
user1231517
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Solving an Irregular Quadrilateral defined by a Circle intersecting with 2 Concentric Circles: their 2 Centers & the 2 Intersection Points
The Problem:
Basically, just what the title says. I have two concentric circles, and a third circle with its center vertically aligned with their center and positioned below it. An irregular ...
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Show that triangle $ABC$ is equilateral
Question
On the small arc $BC$ of the circumscribed circle of the triangle $ABC$, consider two distinct points $M$ and $N$, different from the ends of the arc. We know that the relations $MB+ MC=MA$ ...
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How to show that two certain chords of a circle passing through the incentre of a given triangle are equal?
Let $I$ be the incenter of a triangle $\triangle ABC$. The circle $AIB$ meets the sides $BC$ and $AC$ at points $M$ and $N$, respectively. I'm trying to prove then that $BM=AN$. Here's a figure for ...
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Edges of a $K_4$ cannot be too short
This problem arised while I was working through Erdös-Bollobas's solution for Ramsey-Turán Problem however i believe that details for that isn't necessary.
Consider the following construction
\begin{...
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What is a gyrational square in this context?
This is a diagram of quadrilaterals and their duals. Within this diagram, there is a square (shown with 8 lines of symmetry) and right below that there is a "gyrational square" shown with no ...
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Proving the relation between sides and diagonals of parallelogram without trigonometry and Pythagoras theorem
I am looking for a way to prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides. However, I would like to know whether it is possible to ...
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Angle chasing problem with quadrilateral - possible approaches?
CONTEXT
I was recently working on some Langley-style problems, and wanted to construct some others, based on the "reverse engineering" approach I developed to solve them.
PROBLEM
The ...
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Angle trisection and perimeter of inscribed quadrilateral
I am having trouble solving the following highschool textbook exercise:
Let $AM$ and $AN$ be two chords in a circle of radius $r$ having the same length and such that the sine of the acute angle $\...
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Is satisfying the secant property a sufficient condition for cyclicity of a quadrilateral?
You have a cyclic quadrilateral $ABCD$ with diagonals $AC$ and $BD$ meeting at $P$.
If $PA \cdot PC = PB \cdot PD$, does this necessarily imply $ABCD$ is cyclic? I was unable to find any statement of ...
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Proving the quadrilateral is cyclic [closed]
How does one prove that the quadrilateral $DFPI$ is cyclic given that the greens angles are congruent? This is a solution to a larger problem, however I do not understand this step when they deduce ...
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Show that the quadrilateral $BCED$ is inscribed...
Problem
The triangle $ABC$, with $AB>AC$ and $\angle A \neq 90$, is inscribed with a center circle $O$, and $T$ is the diametrical point opposite $A$. The tangent in $T$ to the circle intersects ...
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What does adjacent similar triangles mean?
PROBLEM
In a convex quadrilateral $ABCD$, any triangle determined by a side and the point of intersection of the diagonals has the same area as the adjacent similar triangles. Show that the ...
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On the products of lengths of adjacent sides in a cyclic quadrilateral
I'm trying to fill in the details of a proof for which I only have a figure, as shown below:
...and a claim that refers to this figure1:
$$
\overline{EG} : \overline{GF} = ΔEDC : ΔFCD = \overline{ED}·...
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Equation of a triangle. Equation of a quadrilateral. Book reference
Analytic geometry textbooks usually teach the equation of a straight line and the equations of a circle and conic sections (equation of an ellipse/hyperbola/parabola). None (except a russian one) as ...
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Angle Chasing in a convex quadrilateral
Find the measures of the angles of the convex quadrilateral ABCD, if $\angle ACD = 78°$ , $\angle BDC = 22°$, $\angle CBD = 12°$ and $\angle CAD = 24°$.
Source: Romanian Mathematical Gazette
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Proving that a Quadrilateral is cyclic made by intersection of perpendiculars from $B,C,D$ from $\triangle ABC$ where $D$ is a pont on $BC$ .
(sorry, couldn't come up with a better title)
The question:
$D$ is a point in the base $BC$ of $\triangle ABC$ and through $B, D, C$ lines are drawn perpendicular to $AB,AD,AC$ respectively meeting ...
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Let ABCD be a convex quadrilateral with AD = BC. Show that AD and BC determine congruent angles with the line passing the midpoints.
Let ABCD be a convex quadrilateral with AD = BC. Show that AD and BC determine congruent angles with the line passing through the midpoints of sides AB and CD...
MY IDEAS
MY DRAWING
As you can see i ...
user1104319
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How prove that $AD$ and $BC$ of the given convex quadrilateral are parallel?
Here is the problem:
In the convex quadrilateral $ABCD$, it is known that $AD > BC$, points $E$ and $F$ are the midpoints of the diagonals $AC$ and $BD$, respectively, $EF = \frac{1}{2}(AD - BC)$. ...
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Let $C_1(O_1, r)$ and $C_2(O_2, r)$ be two congruent secant circles. Calculate in function of the $r$, the area of the quadrilateral $AO_1BO_2$.
Let $C_1(O_1, r)$ and $C_2(O_2, r)$ be two congruent secant circles, so that $C_1 \cap C_2$ = {A, B} and $O_1 \in C_2, O_2 \in C_1$. Calculate in function of the radius, the area of the quadrilateral ...
user1104319
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Solution explanation for finding the value of Angle $\theta$. [duplicate]
Given the quadrilateral $ABGE$, angles $ABG$ and $EAB$ are both $80$ $degrees$, angle $GAB$ is $50$ $degrees$, and angle $EBA$ is $60$ $degrees$. Find the value of angle $\theta$.
I use the solution ...
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A geometry problem from an Italian contest
Consider a quadrilateral $ABCD$ in which the properties listed in the figure below are given.
We want to calculate the length of $BC.$
Using the first and two of the three angle properties it's easy ...
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A convex quadrilateral causes minimal total distance to vertices at the intersection of diagonals
Given a convex quadrilateral, prove the point on the plane with minimal total distance to the four vertices (that is, summing each of the four distances) is the intersection of the diagonals.
This ...
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Calculating the area of a quadrilateral which circumscribes a circle and the distance of a point lying inside it to all the vertices is given. [closed]
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 $ABCD$.
I've ...
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Barycentric coordinates of the origin point of a quadrilateral [closed]
I have a quadrilateral formed by the points A, B, C, D.
I want to get the barycentric coordinates u, v, w, x so that:
Au + Bv + Cw + Dx = [0, 0]
How do I find the values of u, v, w, x knowing the ...
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Geometry Problem with Isosceles Triangles and Cyclic Quadrilaterals
PQR is a right-angled triangle at P and has PQ<PR. Point T is on QR so PQ=QT. Point O is the midpoint of PT. Let X be the point on the circumference of triangle PTR so that angle PXQ=90. Prove that ...
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Find the angle between the diagonals in a quadrilateral [closed]
Find the missing angle, $\theta$, between the diagonals in a quadrilateral.
The angle $\angle ABC$ is right.
I can find any other angle, except the angles between the diagonals. Any hint of how to ...
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Why is $x$ not equal to $75$ degrees?
Someone told me,
Draw $B C$. Let $\angle D B C=y$. Then $30+2 y=180 \Rightarrow y=75$. Now, notice
that ABCD is a cyclic quadrilateral. As a result, $x+y=180 \Rightarrow x=105$ but
I cannot see ...
user685056
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Find an unknown angle in a quadrilateral intersecting with a circle [closed]
I have tried to connect $GO$, $HO$, $IO$, $JO$, $EO$, $AO$, and $FO$, and I know that since the chords are equal, and so as its angles at the center. However, I do not know how to proceed to find the ...
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Prove $EB=EC$ and that $F,M,G,C$ are concyclic in the given figure
Given is a quadrilateral $ABCD$ in which $\angle DAB=\angle CDA=90$. Point M is the midpoint of side $BC$ and circumscribed circles of triangles $\triangle ABM$ and $\triangle DCM$ meet at points $M$ ...
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Does adding a point to a triangle make it a quadrilateral? and if so can a triangle be any polygon?
Does adding a point to a triangle make it a quadrilateral? and if so can a triangle be any polygon?
does the next triangle a quadrilateral?
Or, does a point have to change a side angle?
Thank you for ...
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How to solve for missing angle? (Triangles)
I was looking around at geometry questions and came across this
Seemed simple but I got stuck after filling in everything I knew. This is what I couldn’t get passed
Only thing I could think of doing ...
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Quadrilateral with 2 known coordinates, one known edge vector and 4 known lengths, what are the missing coordinates?
I'm doing some generative design for architecture and I can't quite get my high school geometry over the line on this one. I have filled pages with pythagoras; I must be ignorant of a more powerful ...
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Is my proof (that the summit angles of a Saccheri quadrilateral are congruent) valid?
In Greenberg's (2008) textbook, the following proof that the summit angles of a Saccheri quadrilateral are congruent is given on p. 178.
By hypothesis and SAS, $\triangle DBA \cong \triangle CAB$. ...
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Area of rhombus formed at the intersection of 2 circles
Consider 2 circles with radius r. Question is to find the area of the rhombus formed by joining the intersections. The edges of rhombus are the centers of the 2 circles and the 2 intersecting points.
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Prove that EFZY is a cyclic quadrilateral
The incircle of $\Delta ABC$ touches $BC$, $CA$, and $AB$ at $D$, $E$, and $F$ respectively. $X$ is a point inside $\Delta ABC$ such that the incircle of $\Delta XBC$ touches $BC$ at $D$ also, and ...
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Fastest way to find largest inscribed circle in convex quadrilateral?
I'm looking for the numerically fastest algorithm to determine coordinates & radius of the largest inscribed circle of a convex quadrilateral.
I've found plenty of input on approaches for general ...
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Looking at a possible inconsistency in the areas of two right trapezoids forming a larger one
Could anybody explain to me why the difference arises below?
Area of right trapezoid for the total figure, i.e. $AEFD$, is $131978$
For trapezoid $ABCD$ the area is $62196$
For trapezoid $BEFC$ the ...
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