Questions tagged [quadrilateral]

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

Filter by
Sorted by
Tagged with
0 votes
1 answer
28 views

Angles of a Quadrilateral

I have a quadrilateral ABCD. If I know the angles $DAB, DCB, ADB, CDB, ABD, CBD$, can I obtain the angle $DAC$? I know that I can do it by some coordinate geometry or the sine formula, but is there a ...
user avatar
2 votes
2 answers
69 views

Which quadrilaterals are possible as the intersection of a plane and a rectangular pyramid?

Suppose I have a pyramid with a rectangular base, and consider a plane that does not intersect the base or the apex (it only intersects the four faces incident to the apex). When the plane is parallel ...
user avatar
  • 2,053
0 votes
0 answers
25 views

Property about complete cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral inscribed in a circle of center $O$ and let $AB\cap DC=\{E\}$, $AD\cap BC=\{F\}$ and $AC\cap BD=\{X\}$. Let $OE\cap FX=\{G\}$ and prove that quadrilaterals $AOGB$ ...
user avatar
  • 758
0 votes
1 answer
70 views

Circumcenter of acute-$\Delta ABC$ is $O$. Line $AC$ intersects circumcircle of $\Delta AOB$ at point $X$, in addition to vertex $A$. Prove $XO⟂BC$.

PROBLEM The circumcenter of an acute-$\mathtt{\Delta ABC}$ is $\mathtt{O}$. Line $\mathtt{AC}$ intersects the circumcircle of the $\mathtt{\Delta AOB}$ at a point $\mathtt{X}$, in addition to the ...
user avatar
  • 107
0 votes
0 answers
22 views

What is the formula to find the quadrilateral inner quadrilateral side length?

Sorry if it is a silly question. I want to find an inner quadrilateral area for that I need one of the inner quadrilateral lengths. I don't know the formula if you can please help me. Lengths of the ...
user avatar
  • 109
1 vote
2 answers
45 views

Given rectangle $ABCD$ with $K$ the midpoint $AD$ and $AD/AB=\sqrt{2}$, find the angle between $BK$ and diagonal $AC$.

Could someone help me with this little problem of geometry of quadrilaterals? In rectangle $ABCD$, let $K$ be the midpoint of side $AD$. If we know that $AD/AB =\sqrt{2}$, find the angle between $BK$ ...
user avatar
1 vote
1 answer
42 views

Tangential Equidiagonal (Irregular) Quadrilateral with integer coordinates

Playing with mathematics this week-end, I ended up with this question I couldn't solve, so I'm asking here. I try to find a Quadrilateral ABCD with integers (cartesian) coordinates (points are on a ...
user avatar
  • 9,970
3 votes
4 answers
140 views

Angles of two connected convex quadrilaterals, all lengths given

Consider two convex quadrilaterals sharing one edge with lines meeting at the endpoints of this edge being collinear, so that the seven edges form another, larger convex quadrilateral (see picture). ...
user avatar
  • 31
0 votes
0 answers
54 views

How to construct the centroid of a quadrilateral

I know how to construct the centroid of a quadrilateral as mentioned here. But my question is different from that. We know that if points B,C,G are given in geometry plane and for locating point A ...
user avatar
1 vote
1 answer
30 views

Find the perimeter of a parallelogram given angle bisectors

Got stumped on another problem when I was tutoring a student in Geometry. Here it is: Given: $ABCD$ is a parallelogram, $\overline{AM},\overline{BN}$ angle bisectors, $DM=4\,\text{ft.}$, $MN=3\,\text{...
user avatar
0 votes
2 answers
50 views

Is this converse results of Varignon's theorem known?

The Varignon's Theorem on Quadrilateral is very well known results of Plane Geometry and we have find the Converse of this theorem on Quadrilateral and generalise this for 2n-sided convex irregular ...
user avatar
2 votes
0 answers
46 views

Relation between longest diagonal and shortest side in convex quadrilateral

Given a quadrilateral with side lengths $a,b,c,d$ and diagonal lengths $e,f$, let us use the shortcut $s=\min\{a,b,c,d\}$ (”smallest side”) and $\delta=\max\{e,f\}$ (”largest diagonal”). My questions ...
user avatar
  • 1,518
-2 votes
1 answer
30 views

In triangle ABC point M lays on the BC such that CM/MB=3/2 [closed]

In triangle ABC point M lays on the BC such that CM/MB=3/2 and point N lays on the AB. AM and CN intersect at point O. AO/OM=5:1. what is the area of the triangle ABC if NBMO quadrilateral's area is ...
user avatar
3 votes
1 answer
127 views

Prove that ratio is $k=2/5$

Let $E , G , F , H$ be the midpoints of sides $AB , BC , CD , DA ,$ respectively of quadrilateral $ABCD.$ The common points of segments $AF , BH,CE,DG $ divide each of into three parts, as shown in ...
user avatar
7 votes
1 answer
253 views

Prove that $ABCD$ is a parallelogram.

Let $E , G , F , H$ be the midpoints of sides $AB , BC , CD , DA ,$ respectively of quadrilateral $ABCD.$ The common points of segments $AF , BH,CE,DG $ divide each of into three parts, as shown in ...
user avatar
1 vote
0 answers
19 views

Are there versions of the Intersecting chords/secants theorem that work on non-cyclic quadrilaterals?

The Intersecting chords theorem The Intersecting secants theorem You know how the Cosine rule is a generalisation of Pythagoras' theorem in that it works for all triangles- not just right-angled ones. ...
user avatar
2 votes
1 answer
109 views

A proposition concerning the position of Pascal points.

While answering a recent question I have found algebraically the following nice proposition related to Pascal's theorem. Let $ABCD$ be a cyclic quadrilateral and let $E$ and $F$ be arbitrary ...
user avatar
  • 23.6k
0 votes
0 answers
41 views

What is the correct definition of a trapezium?

There are two different definitions of trapezium in two different textbooks or resources . Some say that :- Trapezium is a quadrilateral with atleast one pair of opposite sides parallel . And the ...
user avatar
  • 1
1 vote
1 answer
48 views

Parallelogram and side lengths

Using the diagram, find $x$ and $y$ if $ABCD$ is a parallelogram. Firstly, we can conclude that $$\measuredangle BAC=\measuredangle ACD=25^\circ \text{ (alternate angles)}$$ Then in triangle $ACD$ we ...
user avatar
  • 1,903
3 votes
1 answer
77 views

Property of bicentric quadrilateral involving the Miquel point of $OMXN$

Let $ABCD$ be a bicentric quadrilateral, let $O$ be its circumcenter and $I$, its incenter. Let $M$ and $N$ be the midpoints of $AC$ and $BD$ repectively. Let $X$ be the intersection of $AC$ and $BD$. ...
user avatar
  • 758
3 votes
1 answer
77 views

Given the length of the diagonals, what is the maximum perimeter of a quadrilateral?

Suppose the length of the diagonal $d$. I'd like to maximize the perimeter of a quadrilateral containing two of these diagonals. I know that for a rectangle instead of a quadrilateral, the solution is ...
user avatar
0 votes
0 answers
41 views

Classify the nature of a quadrilateral determined by the coordinate axes and lines $y=\frac{3}{5}x+\frac{4}{5}$ and $y=\frac{-5}{3}x+\frac{16}{3}$

I have a problem in determining the nature of a quadrilateral. This is generated by the intersection of two straight lines: $$y=\frac{3}{5}x+\frac{4}{5}\qquad y=\frac{-5}{3}x+\frac{16}{3}$$ Here the ...
user avatar
  • 320
0 votes
0 answers
28 views

What does the little arrow mean between two segments of a Quadrilateral

in the below image, what does the little arrow between AD and BC mean? Thanks!
user avatar
  • 9
0 votes
0 answers
21 views

Derivation of area in cyclic quadrilateral that involves the circumradius

So there are two formulas to obtain the area in cyclic quadrilaterals. The first is similar to Heron's formula and I was able to derive this using trigonometric identity. $$ A = \sqrt{(s-a)(s-b)(s-c)(...
user avatar
2 votes
0 answers
72 views

Isometric axis problem : prove that $OS\perp PQ$

Let quadrilateral $ABCD$ be inscribed in $(O)$. Let $I_1, I_2, I_3, I_4$ be the center of the circle inscribed in triangles $ABD, ADC, DBC, ABC$ respectively. $I_1I_3$ cuts $AC$ at$ P$, $I_2I_4$ cuts $...
user avatar
  • 1,130
0 votes
0 answers
61 views

Point of tangency between a sphere and a quadrilateral

I've got a bit of a problem. My problem is : on a sphere, show that if you have a quadrilateral from space tangent to it, the points of tangency (or the points of intersection between the sphere and ...
user avatar
2 votes
1 answer
95 views

In a quadrilateral $ABCD$, $AB=10$, $BC=33$, $CD=10$ and $DA=15$. If $BD$ is an integer, then $BD=?$

In a quadrilateral $ABCD$, $AB=10$, $BC=33$, $CD=10$ and $DA=15$. If $BD$ is an integer, then $BD=?$ I was not able to do anything in this question. The quadrilateral has not been given cyclic, so we ...
user avatar
  • 719
0 votes
1 answer
93 views

A cyclic quadrilateral and a few tangent circles

Let $ABCD$ be a cyclic quadrilateral and denote by $\mathcal C$ its circumscribed circle. Denote by $P$ the intersection of the lines $(AB)$ and $(CD)$ ; and denote by $Q$ the intersection of the ...
user avatar
  • 4,579
-2 votes
1 answer
54 views

Parallelogram and vectors

In the parallelogram $ABCD$ points $M$ and $N$ are the midpoints of $BC$ and $CD$, respectively. Point $P$ is such that $AMPN$ is a parallelogram. Show that $C\in AP$. I am supposed to use vectors. ...
user avatar
  • 55
1 vote
1 answer
59 views

Lower and upper bound for the area of a quadrilateral

Giving sides $a, b, c, d$ is not enough to obtain the area of a quadrilateral because there are many quadrilaterals possible with these 4 lengths. Question: are there well-known lower and upper bound ...
user avatar
  • 1,597
3 votes
3 answers
173 views

Show that formula for area of rectangle does not depend on unit shape

Whenever I search for some kind of reasoning behind the formula for area of rectangle(length$*$width) it is introduced by counting unit squares. I have recently asked a question where my understanding ...
user avatar
0 votes
0 answers
60 views

For a quadrilateral, if 2 adjacent sides are equal, and 2 opposite angles are 90 degrees, can we say that the quadrilateral is a parallelogram?

For a quadrilateral, if 2 adjacent sides are equal, and 2 opposite angles are equal (90 degrees), can we say that the quadrilateral is a parallelogram ? I feel intuitively it should, but is there any ...
user avatar
0 votes
1 answer
62 views

How to find circle radius from cyclic trapezium

I'm solving some of my countries olympiad problems, and after manipulating all of the data I was able to get the title of the question. I have an isosceles trapezium $ABCD$ which has all of its ...
user avatar
1 vote
1 answer
84 views

Circle geometry sketching and proving

Let K and L be circles with centres O and P, respectively, each of which is exterior to the other circle. Suppose K and L intersect at points A and D, and that the diameter AOB of K intersects L at a ...
user avatar
0 votes
0 answers
63 views

References for Euclidean geometry of quadrilateral

I'm quite familiar with the results in Euclidean geometry of triangles. However, it's not the case with quadrilateral. Is there any sources to learn and to check the complicated, advanced results in ...
user avatar
0 votes
3 answers
78 views

How to find the area of a quadrilateral with all sides different?

How to find the area of a quadrilateral in which the length of all the sides are different. For example one pair of opposite sides are 630 foot and 357 foot, and another pair of opposite sides are 587 ...
user avatar
  • 1,174
6 votes
1 answer
848 views

Find fourth side of quadrilateral given three sides and two angles

To completely determine a quadrilateral, you have to have five independent pieces of information, of sides and angles. If you have five data (all outer sides and a diagonal), finding the angles is ...
user avatar
-2 votes
1 answer
137 views

How to find the angle between diagonals of a quadrilateral when all its angles is known?

I need to find the red angles: $\angle(AOB)$, $\angle(BOC)$, $\angle(COD)$, $\angle(AOD)$. The green angles are known: $\angle(ABC)$, $\angle(BCD)$, $\angle(CDA)$, $\angle(DAB)$. Is there any general ...
user avatar
2 votes
2 answers
59 views

What is the measure of $\measuredangle B$ on the trapeze below?

For reference: In the ABCD trapeze(CB//AD), $m\measuredangle B = 4m \measuredangle D$ and $11AB + 5BC = 5AD$. Calculate $\measuredangle D$. (answer $26,5^\circ$} My progress: I could only find a ...
user avatar
  • 4,019
0 votes
1 answer
224 views

Why the sum of interior angles of any concave or convex polygon is 180×(n-2)°? How to prove this?

I have just read about the sum of interior angles of convex polygons with n sides, which is $$(n-2) × 180°$$ Then I tried to find the sum of interior angles of some concave polygons. Surprisingly, it ...
user avatar
  • 171
2 votes
2 answers
172 views

How to prove in figure that $x=30^\circ$, where we know angles $36^\circ$, $24^\circ$ and $18^\circ$?

I know how to prove $\sin(156^\circ−2x)\sin24^\circ\sin18^\circ\sin36^\circ = \sin(x)\sin(x)\sin(138^\circ−x)\sin(x−12^\circ) $ but if is it possible to prove without trigonometry?
user avatar
  • 6,157
2 votes
0 answers
86 views

Cyclic quadrilateral: opposite angles add up to $180^\circ\Leftrightarrow$ angles in same segments are equal?

There are two well-known results in cyclic quadrilaterals: (a) Opposite angles add upto $180^\circ$. (b) Angles in the same segments are equal. The converse of these two results are also true. The ...
user avatar
2 votes
2 answers
105 views

What is the measure of the $\angle MAD$ in the parallelogram below if $\angle B=110^\circ$

For reference: The perpendicular bisectors of sides $AD$ and $CD$ of an $ABCD$ parallelogram intersect at a point $M$ that belongs to $BC$. Find $\angle MAD$ if $\angle B = 110^\circ $ . My ...
user avatar
  • 4,019
4 votes
2 answers
122 views

Area of Cyclic Quadrilateral

Let $ABCD$ be a quadrilateral inscribed in a circle with diameter $AC$, and let $E$ be the foot of perpendicular from $D$ onto $AB$. If $AD=DC$ and the area of quadrilateral $ABCD$ is $24$, find $DE$. ...
user avatar
  • 435
0 votes
1 answer
34 views

Area of quadrilateral given one diagonal and a line joining midpoints of two adjascene sides

Find the area of the quadrilateral ABCD given $EA=BE$ and $BP=PC$ and $PE\perp BD$ $EA=BE$ and $BP=PC$ $\implies \frac{BE}{BP}=\frac{EA}{PC} \implies PE\parallel AC$ $\implies AC\perp BD$ $\implies $...
user avatar
  • 6,789
0 votes
1 answer
62 views

Irregular convex quadrilateral: Find the diagonal length given all sides and one angle

I stumbled into an Irregular convex quadrilateral in one of my projects and I cannot figure out if the diagonal |BD| can be found. Problem: I have an Irregular convex quadrilateral ABCD defined by the ...
user avatar
1 vote
1 answer
108 views

Find the angle BEC in this quadrilateral - where have I gone wrong?

Question : I have received this question as a challenge from my teacher in Year 8 and I seem to make no progress. All I know is that $DC$ is parallel to $AB$, and that $DA$ is parallel to $AB$ ...
user avatar
2 votes
1 answer
180 views

Angle chasing problem: Find $\angle Q_{2024}Q_{2025}P_{2025}$ in the quadrilateral.

$ABCD$ is a convex quadrilateral where $BC = CD$, $AC = AD$, $\angle BCD = 96^\circ$ and $\angle ACD = 69^\circ$. Set $P_0 = A, Q_0 = B$ respectively. We inductively define $P_{n+1}$ to be the center ...
user avatar
  • 3,823
1 vote
0 answers
61 views

Do the rectangles that circumscribe and share edges with a rhombus have identical dimensions?

I am going to make an app for wrapping presents but I don’t know a lot about useful properties of rhombi and associated rectangles. Given the known dimensions necessary to create a rhombus that ...
user avatar
  • 111
3 votes
2 answers
426 views

Quadrilateral in which diagonal is partially outside?

I just want an example of a quadrilateral in which a diagonal lies "partially outside" the quadrilateral. By that, I mean that some part of a diagonal must be outside and some part inside ...
user avatar
  • 1,235

1
2 3 4 5
12