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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ... 61 views

### 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, $EF = \frac{1}{2}(AD - BC)$. ...
1 vote
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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 ... 37 views

### 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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ... 80 views

### 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 ...
1 vote
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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. ...
1 vote
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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 ...
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 ...