Questions tagged [quadrilateral]

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

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2
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5answers
53 views

Prove that $\frac{AB}{AE} + \frac{AD}{AG} = \frac{AC}{AF}$ in parallelogram $ABCD$, where $E$, $F$, $G$ are points on a line intersecting the sides

Let $ABCD$ be a parallelogram. A line meets segments $AB$, $AC$, $AD$ at points $E$, $F$, $G$, respectively. Prove that $\frac{AB}{AE} + \frac{AD}{AG} = \frac{AC}{AF}$. So recently I've been assigned ...
-3
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0answers
31 views

Kites and Trapeziums and Circle

Let there be a circle with center O and let A, B, C, D, be points on the circle such that O is in the interior of quadrilateral ABCD. Construct the four tangents at A, B, C, D. Let P be the ...
0
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1answer
39 views

A geometry question on Menelaus's theorem | Prove that HF/HC = FG/CG

Let $ABC$ be a triangle. $D, E, F, G, H$ are points such as $E~\in AC$, $D ~\in BC$, $F = AD\cap BE$, $G = \overleftrightarrow{CF} \cap AB$ and $H = ED \cap CG$. Prove that $\dfrac{HF}{HC} = \dfrac{...
1
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2answers
53 views

Prove that $\frac{PQ}{MN} = \frac{|[BCE] - [ADE]|}{[ABCD]}$ in a quadrilateral ABCD where P and Q are related to the diagonals

I've recently been given a few challenge problems that I really want to find out. But for the most part, I just can't figure out how to completely prove the problems. Now one of the problems goes ...
1
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1answer
29 views

Is quadrilateral with two equal opposite sides and joining mid points of other sides divide equally?

Let $ABCD$ a convex quadrilateral such that $AB=CD$. Let $P$ and $Q$ are the mid points of the sides $BC$ and $AD$ respectively. Now if we joint $PQ$, is it divide the quadrilateral in equal area? To ...
1
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1answer
53 views

Area of Triangle inside a Rectangle

Rectangle $WXYZ$ has an area of $25$. Point $U\ \&\ V$ lie at the sides $XY\ \&\ YZ$,respectively$. $$\triangle WXU$ has an area of $6$ & $\triangle WZV$ has an area of $5$. Find the area ...
0
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1answer
22 views

Express 2 vectors in terms of 2 other vectors.

The point P lies on the circle through the vertices of a rectangle QRST. The point X on the diagonal QS is such that $\overrightarrow {QX}$ = $2\overrightarrow {XS}$. Express $\overrightarrow {PX}$, $\...
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1answer
16 views

Area of irregular quadrilateral with diagonals proof

I have an irregular convex quadrilateral with diagonals d and D. These diagonals form an acute angle $\alpha$. I know that I can find the area of this quadrilateral by using this formula: $A = \frac{D\...
1
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1answer
52 views

Problem involving three cyclic quadrilaterals

I found out the following configuration in elementary geometry. I know it is true (by drawing in Geogebra) but I haven't proved it yet. Let $ABC$ be a triangle with the circumcircle $(O)$. $M$ is an ...
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3answers
88 views

$a,b,c,d$ are complex numbers corresponding to points $A,B,C,D$ lying on a circle with origin as center,and chord $AB⟂CD$. Find $ab+cd$

Question Let on the Argand plane $a,b,c$ and $d$ represent the complex numbers corresponding to the points $A,B,C$ and $D$ respectively, all of which lie on a circle having center at the origin. The ...
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1answer
24 views

If $AB = a$, $BC = b$, $CD = c$, $DA = d$ of convex quad, determine the length of the segment $PQ$ from incircles

So recently I've found an interesting problem that I've wanted to solve. Here's the problem: Define $ABCD$ as a convex quadrilateral. The incircle of triangle $BCD$ touches line $BD$ at $P$, and the ...
4
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2answers
221 views

Show this quad is cyclic

Doesn't seem hard but it got me stuck: $I$ is the incenter of $\triangle ABC$ $D$ the contact point of the incircle with $BC$ $M,M'$ are the intersection of the circumcircle of $\triangle ABC$ with ...
0
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1answer
41 views

Prove that $MN=\frac{1}{2}(AB+BC+CA)$

As I was doing a question on quadrilateral, I found out this one, which I couldn't find to relate with any properties of a quadrilateral. It seems there is a trapezium in between, but it's quite ...
6
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2answers
169 views

In quadrilateral $ABCD$, $\angle BAC=\angle CAD=2\,\angle ACD=40^\circ$ and $\angle ACB=70^\circ$. Find $\angle ADB$.

Let quadrilateral $ABCD$ satisfy $\angle BAC = \angle CAD = 2\,\angle ACD = 40^\circ$ and $\angle ACB = 70^\circ$. Find $\angle ADB$. What I tried Ceva’s Theorem (Trigonometry version) Try to ...
5
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0answers
365 views

Generalization of two formulae and an alternative proof of Bretschneider's formula

Below I present two seemingly unknown identities that I then use to provide an alternative proof of Bretschneider's formula. Made the necessary adjustments, the identities can also be used to provide ...
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2answers
26 views

Quadrilateral inside a polygon having no common side,different approach.

Question: Find the number of quadrilaterals formed by joining the vertices of a decagon,that share no common side with the decagon. Approach: Select one vertex first:-This can be done in $10\choose1$=...
0
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1answer
38 views

A geometry problem about areas and lengths. Squares and polygons.

An area inequality: Let $ ABCD $ be a square, $ AB = 1 $. Let's consider a 4-sided polygon $ EFGH $, with a peak on each side of the square. The area of EFGH equals $ \frac{1}{2} $. Prove that it ...
3
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1answer
74 views

how to prove that the segment $IF=HF+GF$ [closed]

$AE$ and $CD$ are the angle bisectors of $\triangle ABC$. $F$ is an arbitrary point on line $DE$. Prove that $GF+HF=IF$. I noticed $3$ cyclic quadrilaterals. Any ideas. Here is the picture
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1answer
36 views

Diagonals of a parallelogram

The diagonals of a parallelogram are $8$ and $4$. They meet at $60°$. Find the sides and area of the parallelogram. I tried to use the cosine rule here having $4$ and $2$ as the sides and $60°$ as ...
2
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1answer
88 views

Inscribed Circles in a Quadrilateral

I found this problem online, where it was asked to prove EF = GH. I was able to prove that, but got intrigued by how the four smaller inscribed circles could be constructed in the first place. That ...
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2answers
61 views

Don't understand the solution of this geometry problem

In this question, AF cuts the parallelogram ABCD. BE, CF, DG are $\perp$s from the other vertices to AF. LM is also a $\perp$ where L is the intersection of 2 diagonals in the parallelogram. Prove CF =...
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0answers
13 views

Prove that the midpoints of a line segment and the mid points of the diagonals of a quadrilateral are concurrent [duplicate]

In quadrilateral $ABCD$, let $AB$ and $CD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Then prove that the midpoints of $AC$, $BD$ and $EF$ are collinear. I cannot make any progress with this question. ...
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1answer
45 views

How to solve for the area for quadrilaterals like these?

How do I solve for the area of the yellow quadrilateral? This question has been bugging me and I haven't been able to figure out much given the areas of the triangles, 2 and 3 and the quadrilateral at ...
1
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1answer
54 views

Show that the area of a triangle is equal to one fourth of a trapezium [closed]

$ABCD$ is a trapezium with $AD || BC$; $AD=3BC$ and a transversal $XY$ cuts $BC$ at $X$ and $AD$ at $Y$. If $EF$ is a line segment contained in $XY$ such that $AE||DF$ . $BE||CF$ and $AE/DF=CF/BE=2$, ...
0
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1answer
32 views

Proofs through transformations

I was given the following question: Polygon ABCD is a kite, with point E as the center and $\bar{AC}$ and $\bar {BD}$ as the diagonals. Which transformations can be used to proove that $\triangle ABC$...
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2answers
61 views

Centroid of a quadrilateral $ABCD$: $\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} + \overrightarrow{GD} = \overrightarrow{0}$

Let $ABCD$ be a quadrilateral and let $G_1$, $G_2$, $G_3$, $G_4$ be he centroids of triangles $ABC$, $BCD$, $CDA$ and $DAB$, respectively. Assume without proof that $G_1G_3$ and $G_2G_4$ always ...
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2answers
63 views

A problem about a quadrilateral and diagonals in Kiselev's Geometry (Exercise 521).

The problem is from Kiselev's Geometry exercise 521: In a quadrilateral $ABCD$, through the midpoint of the diagonal $BD$, the line parallel to the diagonal $AC$ is drawn. Suppose that this line ...
1
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1answer
37 views

Area of Inscribed (Cyclic) Quadrilateral

Let $ABCD$ be an inscribed (cyclic) quadrilateral with $\widehat{BAC}$ $\equiv $ $\widehat{DAC}$. Prove that the area of $ABCD$ is equal to $\dfrac{1}{2}AC^2\sin A$. https://www.geogebra.org/...
2
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2answers
72 views

How to prove that $ABCD$ is a parallelogram?

Let $ABCD$ be a quadrilateral. Let $E$ and $F$ be midpoint of $AB$ and $BC$. The lines $DE$ and $DF$ intersect $AC$ at $M$ and $N$ respectively. Suppose that $AM$ $=$ $MN$ $=$ $MC$. Prove that $ABCD$ ...
1
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2answers
101 views

Making the area of the quadrilateral and the area of a triangle the same

$ABCD$ is a quadrilateral and $X$ is a given point on AD. Find a point Y in AB such that the area of the $\triangle AXY$ is equal to that of $ABCD$. Hence show how to divide the quadrilateral $ABCD$ ...
1
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2answers
49 views

Points $A (b, 2c), B(4 b, 3c), C (5b, c)$, and $D(2b, 0)$ form a quadrilateral. How would you classify the Quadrilateral and explain your steps?

I have homework - Points $A (b, 2c), B (4 b, 3c), C (5b, c)$, and $D (2b, 0)$ form a quadrilateral. How would you classify the Quadrilateral and explain your steps? But I don't know really how to ...
7
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4answers
297 views

Ratio of area covered by four equilateral triangles in a rectangle

The following puzzle is taken from social media (NuBay Science communication group). It asks to calculate the fraction (ratio) of colored area in the schematic figure below where the four colored ...
0
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1answer
35 views

Property of cyclic quadrilaterals [closed]

Suppose $ABCD$ is a cyclic quadrilateral and $P$ is the intersection of the lines determined by $AB$ and $CD$. Show that $PA·PB= PD·PC$ Could you help me please, I have no idea how to relate the ...
2
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1answer
27 views

Parallelogram Inequality

Let M be a point inside parallelogram ABCD. Then prove that $MA + MB + MC + MD < 2(AB + BC)$ I tried this problem using Triangle Inequality but couldn't proceed. Please help.
1
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1answer
61 views

Finding angles in quadrilateral.

Let $ABCD$ a quadrilateral, $AC\cap BD=\{O\}$, $\angle A=110^{\circ}$, $\angle DOC=60^{\circ}$ and $AC=AB+CD=BD$. Find $\angle B$, $\angle C$. I tried a lot of constructions. I think $ABCD$ is a ...
1
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2answers
99 views

Quadrilateral $ABCD$ with $AB=AD$, $\angle BAD=60^\circ$, $\angle BCD=120^\circ$. Prove $BC+DC=AC$ [closed]

In a given quadrilateral $ABCD$, we have $$AB = AD, \angle BAD = 60^\circ, \angle BCD = 120^\circ$$ Prove that $$ BC + DC = AC$$ I know the quadrilateral is cyclic. I have been able to solve this for ...
0
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1answer
41 views

A convex quadrilateral with sides $a$, $b$, $c$ has maximal area when its fourth side satisfies $x^3-(a^2+b^2+c^2)x-2abc=0$

Three sides of a convex quadrilateral $ABCD$ have lengths $AB = a$, $BC = b$, and $CD = c$. If the area of ​​the quadrilateral is as large as possible, prove that the length $x$ of the fourth side ...
0
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2answers
46 views

Prove that $d^2=a^2+b^2+c^2-2[ab\cos B+bc\cos C+ac\cos(α)]$ [closed]

The angle between the AB and CD sides of an ABCD convex quadrilateral is equal to $\alpha$. Considering that AB = a, BC = b, CD = c, DA = d, prove that: $$d^2=a^2+b^2+c^2-2[ab\cos B+bc\cos C+ac \cos(\...
2
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1answer
36 views

Find the largest diagonal of a parallelogram if the area is known

The original question: In a parallelogram, the length of one diagonal is twice of the other diagonal. If its area is $50\text{ sq. metres}$, then the length of its bigger diagonal is... A) ...
0
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4answers
89 views

Possible criterion for proving parallelogram.

A quadrilateral $ABCD$ has $∠BAD = ∠BCD$ and diagonal $AC$ bisects diagonal $BD$ at $P$. Is it necessarily a parallelogram? If not, give an example of such a quadrilateral. Provide proof. I was ...
1
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2answers
45 views

Is it a Trapezoid?

I have a convex quadrilateral (ABIJ) similar to a trapezoid. By condition $ABCDEFGHIJ$ is a regular decagon. Then $AB = JI$, $\angle BAJ = \angle IJA$. We can prove that $\angle IBA= \angle BIJ, ...
0
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1answer
40 views

Equilateral triangle in a regular pentagon

I don't understand how the line segment of $|BF|$ equals to $|BC| = |FC| = |AB| = |AE|$. How can the line segment maintains the equilateral triangle? The instructor says while drawing that it just ...
0
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1answer
25 views

Relation between areas in a trapezoyd

A trapezoid of ABCD vertices is inscribed in a circle, with radius R, being AB = R and CD = 2R and BC and AD being non-parallel sides. The bisectors of the internal trapezoidal angles, so that the ...
0
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1answer
31 views

$ABCD$ a convex quadrilateral. Show that $MN \le \frac{a+kc}{1+k}$

Let $ABCD$ be a convex quadrilateral and $k \gt 0$. Let $M \in [AD]$, $N \in [BC]$ as $\frac{MA}{MD} = \frac{NB}{NC} = k$. Show that if $AB = a$, $CD = c$, then $MN \le \frac{a+kc}{1+k}$. I really ...
4
votes
5answers
126 views

Find an angle in the given quadrilateral

In the following problem, I want to find the angle marked as $x$. It seems so simple and yet I am out of ideas. It is very easy to get all angles except two of them: angle ADB and angle CBD. Is ...
0
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2answers
34 views

geometry problem dealing with quadrilateral and proving two lines perpendicular to each other and equal in magnitude

On each side of a quadrilateral ABCD,squares are drawn.The centers of the opposite squares are joined.Show that PR and QS are perpendicular to each other and equal in magnitude. pure geometry is ...
2
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1answer
41 views

How would I solve this, considering I have no values whatsoever?

I think that SQ is straight and so have tried to use Pythagoras, which leaves me with $a + b + c = ac/b$, but I don't see any values. How could I find values?
2
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2answers
64 views

Finding the sum of the areas of three squares from the sum of three other quadrilaterals.

The sum of the areas of squares $X,$ $Y,$ and $Z$ is 112. Find the sum of the areas of squares $P,$ $Q,$ and $R.$ How can I form a system of equations to solve for the sum of the areas of squares P, ...
0
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1answer
36 views

Finding the diagonal of a rectangle.

What fraction of the rectangle is shaded? (You may assume that each line, other than the diagonal of the rectangle, is parallel to some side of the rectangle.) Is there a way to solve this without ...
0
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2answers
40 views

Finding length of line that intersects trapezoid diagonals.

In trapezoid $ABCD,$ base $\overline{AB}$ has length 6, and base $\overline{CD}$ has length 18. A line passes through the intersection of the diagonals, parallel to the bases. This line intersects $\...

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