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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 prove opposite angle bisector theorem for convex quadrilaterals?

Let $ABCD$ be a convex quadrilateral with $BL$ and $DL$ be its angle bisectors. I want to know how to prove that the acute angle $\alpha$ between these bisectors is equal to $\frac{\left|\angle A - \...
Rusurano's user avatar
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Problem of angles in a quadrilateral [closed]

Given a quadrilateral ABCD, E is a point on AD. F is a point inside ABCD such that CF, EF bisects ∠ACB and ∠BED respectively. Prove that ∠CFE = 90° + 1/2 (∠CAD + ∠CBE).
Aryan Malik's user avatar
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Prove centre is inside cyclic quadrilateral with perpendicular diagonals

Let $ABCD$ be a convex cyclic quadrilateral, and the diagonals $AC$ and $BD$ are perpendicular. The circumcircle of $ABCD$ has centre $O$. I am trying to prove that the centre $O$ is inside $ABCD$. I ...
wenbang's user avatar
2 votes
1 answer
130 views

Defining an Isosceles Trapezoid

A Trapezoid is a quadrilateral with at least one set of parallel sides. An Isosceles Trapezoid is a Trapezoid where the legs are of equal length. These definitions are called inclusive. This means ...
Suamere's user avatar
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Three sides of a quadrilateral are congruent, and their two included angles are right. Must the quadrilateral be a square?

If quadrilateral $ABCD$ has $\angle ABC = \angle DCB = 90^\circ$ and $AB=BC=CD=4$, must $ABCD$ be a square? Logically, it seems $ABCD$ must be a square, but how can it be proven that $AD$ must be ...
Problem_Solving's user avatar
1 vote
1 answer
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question on geometry of cyclic quadrilaterals [duplicate]

ABCD is a cyclic quadrilateral. AB produced meets DC produced at F. AD produced meets BC produced at E. Prove that the angle bisectors of ∠AEB and ∠AFD meet at right angles. From the diagram we have ...
Aarush Singh's user avatar
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2 answers
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Area of the parallelogram formed by joining the midpoints of the sides of a quadrilateral

$E, G, F$ and $H$ are the mid-points of the sides of the quadrilateral $ABCD$. Prove that the area of $EGFH$ is half of the area of $ABCD$. Since the sides of $EGFH$ are parallel to the diagonal of ...
Etemon's user avatar
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4 votes
4 answers
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I need help in understanding the alternative solution provided to solve this geometry question of calculating area of quadrilateral

Question: Solution provided: I understand this part that equal chords of a circle subtend equal angles at the center, but after this the faculty transformed this whole diagram to one shown below in ...
Vasu Gupta's user avatar
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1 vote
1 answer
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Question on quadrilaterals

The distance between two parallel sides $AB$ and $CD$ of a trapezoid is $12$ units. $AB = 24$ units; $CD = 15$ units. $E$ is the mid-point of $AB$ . $O$ is the point of intersection of $DE$ with $AC$ ....
Aarush Singh's user avatar
2 votes
0 answers
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Prove the area of a quadrilateral $A_1B_1C_1D_1$ has a local minimum at $A_1=A$

$ABCD$ is a cyclic quadrilateral. Its diagonals $AC,BD$ intersect at $P$. Let $E$ be the point on $AB$ such that $AE:EB=\tan\angle BAP:\tan\angle ABP$. Let $F$ be the point on $BC$ such that $BF:FC=\...
hbghlyj's user avatar
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Find the length of segment $AD$ in this trapezoid [closed]

$ABCD$ is a trapezoid with $AB \parallel CD$ and $AB \perp BC$. In addition, the two diagonals $AC$ and $BD$ satisfy $ AC \perp BD$. Segment $BC = 3$. And finally, the area of the blue triangle is ...
Quadrics's user avatar
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1 vote
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Find $x$ in this concyclic quadrilateral

$ABCD$ is a concyclic quadrilateral, with $\angle A = 60^\circ$, and $ AB = 10, BC = x , CD = x+2 , DA = x+4 $. Find $x$. My attempt: Using the vector method, we can express the horizontal and ...
Quadrics's user avatar
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3 votes
3 answers
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Find $x$ in this quadrilateral

A quadrialteral $ABCD$ has $AB = 10$, $\angle A = 50^\circ, \angle B = 120^\circ$, $ BC = x , CD = x + 2 , AD = x + 4 $. Find $x$. My attempt: Applying the law of cosines to $\triangle DAC$ and $\...
Quadrics's user avatar
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14 votes
5 answers
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Let $ABCD$ be a convex quadrilateral. If the measure of the angles $A=90, C=96, D=78$ and $BC=2*AB$, then the measure of the angle $ABD$ is?

The problem Let $ABCD$ be a convex quadrilateral. If the measure of the angles $A=90°, C=96°, D=78°$ and $BC=2*AB$, then the measure of the angle $ABD$ is...? The idea As you can see I calculated ...
IONELA BUCIU's user avatar
2 votes
1 answer
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Maximizing Perimeter of Triangle PDE on a Parabola and Finding Coordinates of N for Rhombus Formation

Given a parabola in the Cartesian plane defined by the equation ( $y = -\frac{1}{2}x^2 + \frac{3}{2}x + 2 $), it intersects the x-axis at points A and B, and the y-axis at point C. Consider a point P ...
Oth S's user avatar
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1 answer
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Reflecting point across bisector of acute angle leads to perpendicular elsewhere

Exercise. Let $\angle BAC$ be an acute angle. In the interior of $\angle BAC$ there is a point $P$ whose projections onto the sides $AB$ and $AC$ are precicely $B$ and $C$. Draw the bisector of the ...
Linear Christmas's user avatar
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Can Parameshvara's formula be proven without trigonometry?

Parameshvara's formula connects sides of cyclic quadrilaterals with its circumradius and area. Ptolemy's theorem can be proven by triangle similarity, as shown here. Brahmagupta's formula can also be ...
Rusurano's user avatar
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1 vote
2 answers
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Finding angles in a convex quadrilateral, given only the lengths and angle between opposing sides

I'm currently struggling with finding the angles of a quadrilateral and can't seem to come up with a solution. This image shows a visual representation of my problem. I am trying to mathematically ...
Kacper Jarzymowski's user avatar
2 votes
1 answer
53 views

Show that the sum of the perimeters of the circles is at most $\pi \sqrt{n}$ (the lines are not allowed to cut the interior of a subpolygon).

Question: Let us divide by straight lines a quadrangle of unit area into $n$ subpolygons and draw a circle into each subpolygon. Show that the sum of the perimeters of the circles is at most $\pi \...
Mods And Staff Are Not Fair's user avatar
1 vote
0 answers
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Does the Ptolemy's inequality hold for concave quadrilaterals?

We used a convex quadrilateral to prove that $\overline{AB}\cdot \overline{CD}+\overline{BC}\cdot \overline{DA} \ge \overline{AC}\cdot \overline{BD}$ is true for non-cyclic quadrilaterals. But I was ...
Bennkaitoshinichi's user avatar
1 vote
1 answer
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Analytical Method for Finding the Closest Point on a 3D Quadrilateral Polygon Face from a Line Segment

I am interested in developing an analytical method to determine the closest point on a convex quadrilateral polygon face, defined by four points (A, B, C, and D), from a given line segment connecting ...
thi's user avatar
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3 votes
3 answers
173 views

Sum of angles in a $1$-by-$3$ rectangle [closed]

This problem was in a competition for a job. It seems simple BUT the challenge is you cannot use trigonometry. Let there be 3 squares with side length of $\ell$ arranged in such a way that it forms a ...
ShrekLover's user avatar
2 votes
1 answer
62 views

An extension of Brahmagupta's theorem.

As we know, there are conjectures that are easy to formulate but difficult to prove, and there are conjectures that are easy to prove but difficult to conceive. This conjecture is simple to conceive ...
George Plousos's user avatar
2 votes
3 answers
67 views

Prove a convex quadrilateral with perpendicular diagonals and one pair of congruent, non-consecutive angles is a kite.

I strongly suspect the conditions in the title are sufficient for a kite, but I am unable to give a proof, direct or otherwise. I've attempted to proceed directly with triangle congruence strategies ...
casualmath's user avatar
2 votes
1 answer
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Determine the angles of quadrilateral that make it concyclic

Inspired by a recent post, consider the following problem. You are given the four sides lengths of a quadrilateral $ABCD$. Let these sides be $a = AB , b = BC, c = CD, d = DA $. I want to determine ...
Quadrics's user avatar
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2 votes
2 answers
150 views

Number of non-congruent quadrilaterals with vertices chosen from among seven points equally distributed on a circle

Seven points are equally distributed on a circle. How many non-congruent quadrilaterals can be drawn with vertices chosen from among the seven points? My approach: If there are $7$ points and we have ...
NOT ACID's user avatar
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Find the inellipse of a convex quadrilateral given its axes direction

In a recent post I've investigated the inellipse of a triangle with the direction of its major axis and major to minor axes ratio specified. In this problem, you're given a convex quadrilateral with ...
Quadrics's user avatar
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1 vote
1 answer
113 views

Geometry problem with medians and areas

Let $ABC$ be a triangle with centroid $G$. A perpendicular line from $G$ to the line $BG$ intersects the parallel through $A$ to the line $BC$ in $D$. Prove that $AC\cdot BD\geq 2\cdot area[AGBD]$. So ...
JohnnyC's user avatar
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2 votes
2 answers
102 views

Constructing a quadrilateral with $3$ equal sides given their midpoints

Construct a quadrilateral with three equal sides, given the midpoint of each of these three sides. Also it is given that which of these three midpoints is for the middle side (a side between the two ...
Etemon's user avatar
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How to calculate the angle of intersection of the diagonals of a quadrilateral? [duplicate]

Let's consider two triangles ABC and DEF. All the six angles of both the triangles are known. And, BC = DE. Now, I form a quadrilateral by combining these triangles such that the side BC or DE is now ...
Motivix's user avatar
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0 votes
1 answer
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How to find the area of the given circle?

Given, we have an $\square ABCD$ with a side length of $1\text{cm}$. We construct its diagonal $AD$. From $C$, we draw the altitude $CE$ of $\triangle ACD$. Now, we construct the altitude $FE$ of the ...
Sambhav Khandelwal's user avatar
2 votes
1 answer
100 views

How can we find the area of the given highlighted figure?

Problem Given, we have a square ▢ABCD with a side of 2cm. We construct the diagonal and draw a circle tangent to all the 3 sides of the triangle △ABD formed. Then, we divide this circle into 4 equal ...
Sambhav Khandelwal's user avatar
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0 answers
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eccentricity of the nine-point conic and the Gergonne-Steiner Conic

I read from here that the nine-point conic (QA-Co1) of a quadrangle $ABCD$ has eccentricity $e$ satisfying $$\frac{{{{({e^2} - 2)}^2}}}{{{e^2} - 1}} = \frac{{{{\sin }^2}(A + C)}}{{\sin A\sin B\sin C\...
hbghlyj's user avatar
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1 vote
2 answers
340 views

Another olympiad problem with a complete quadrilateral

Let $ABCD$ be a convex quadrilateral whose diagonals intersect at $P$. Let $X$ and $Y$ be points such that quadrilaterals $ABPX$, $CDXP$, $BCPY$, and $DAYP$ are cyclic. Lines $AB$ and $CD$ intersect ...
Natrium's user avatar
  • 161
3 votes
1 answer
141 views

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 ...
Lemon's user avatar
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1 vote
2 answers
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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 ...
Jean Marie's user avatar
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2 votes
2 answers
97 views

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-...
aarbee's user avatar
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1 vote
2 answers
83 views

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 ...
Bob Dobbs's user avatar
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1 vote
3 answers
100 views

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 ...
Quadrics's user avatar
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-1 votes
1 answer
60 views

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 $\angle (BAO-OBC)$?
Md. Sayan Khan's user avatar
2 votes
1 answer
96 views

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 $...
hbghlyj's user avatar
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0 votes
1 answer
73 views

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 \...
hbghlyj's user avatar
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0 votes
1 answer
41 views

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 ...
Aanchal Jha's user avatar
1 vote
0 answers
54 views

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 ...
danik0011's user avatar
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0 answers
48 views

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} ...
user avatar
2 votes
1 answer
56 views

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 ...
ChaosMageX's user avatar
1 vote
1 answer
155 views

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 $MB+ MC=MA$ ...
IONELA BUCIU's user avatar
2 votes
4 answers
201 views

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 ...
math-physicist's user avatar
2 votes
1 answer
53 views

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{...
total dependent random choice's user avatar
5 votes
1 answer
257 views

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 ...
Fomalhaut's user avatar
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