Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now

Questions tagged [quadrilateral]

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

Filter by
Sorted by
Tagged with
226
votes
24answers
49k views

Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers

An exam for high school students had the following problem: Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a circle be determined by the points $E$, $B$ and ...
57
votes
14answers
13k views

Area of a square inside a square created by connecting point-opposite midpoint

Square $ABCD$ has area $1cm^2$ and sides of $1cm$ each. $H, F, E, G$ are the midpoints of sides $AD, DC, CB, BA$ respectively. What will the area of the square formed in the middle be? I know that ...
37
votes
9answers
3k views

Area of parallelogram = Area of square. Shear transform

Below the parallelogram is obtained from square by stretching the top side while fixing the bottom. Since area of parallelogram is base times height, both square and parallelogram have the same area. ...
30
votes
9answers
7k views

Is every parallelogram a rectangle ??

Let's say we have a parallelogram $\text{ABCD}$. $\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two parallel lines $\text{AB}$ and $\text{CD}$, So, $$ar\triangle \...
29
votes
3answers
2k views

A conjecture involving prime numbers and parallelograms

As already introduced in this post, given the series of prime numbers greater than $9$, let organize them in four rows, according to their last digit ($1,3,7$ or $9$). The column in which they are ...
23
votes
11answers
6k views

Is there a formula to calculate the area of a trapezoid knowing the length of all its sides?

If all sides: $a, b, c, d$ are known, is there a formula that can calculate the area of a trapezoid? I know this formula for calculating the area of a trapezoid from its two bases and its height: $...
19
votes
2answers
19k views

Maximum area of a square in a triangle

I want to calculate the area of the largest square which can be inscribed in a triangle of sides $a, b, c$ . The "square" which I will refer to, from now on, has all its four vertices on the sides of ...
15
votes
4answers
685 views

Trisect a quadrilateral into a $9$-grid; the middle has $1/9$ the area

Trisect sides of a quadrilateral and connect the points to have nine quadrilaterals, as can be seen in the figure. Prove that the middle quadrilateral area is one ninth of the whole area.
14
votes
4answers
315 views

Angles of triangle $\triangle XYZ$ do not depend on the position of point $P$ (proof needed)

Let $ABCD$ be a fixed convex quadrilateral and $P$ be an arbitrary point. Let $S,T,U,V,K,L$ be the projections of $P$ on $AB,CD,AD,BC,AC,BD$ respectively. Let $X,Y,Z$ be the midpoints of $ST,UV,KL$. ...
10
votes
1answer
283 views

Interesting tiling with a lot of symmetrical shapes

I have such an interesting observation: if I take a square grid and rotate it over itself by atan(3/4) , it forms a structure which has four axes of reflection symmetry: The resulting structure is ...
9
votes
7answers
1k views

What is a simple (or not) way of finding the lengths of the diagonals of this rhombus?

I recently taught a group of geometry students about properties of rhombuses, and gave them a set of homework exercises created by a previous instructor which included the following problem. If a ...
9
votes
4answers
19k views

Why study quadrilaterals?

My niece is in the 10th grade, and they have to do lot of theorems related to quadrilaterals. And, I was surprised to know that they have to learn by rote some theorems. This has made her feel that ...
9
votes
2answers
305 views

Volume of an Irregular Octahedron from edge lengths?

Does anyone know how to calculate the volume of an irregular octahedron from the lengths of the edges? The octahedron has triangular faces, but the only information are the edge lengths. ...
9
votes
1answer
429 views

Point that divides a quadrilateral into four quadrilaterals of equal area

Consider an irregular quadrilateral $ABCD$. Let $E,F,G,H$ be the midpoints of its edges. It seems that there is a point $K$ such that $$ S_{AHKE} = S_{EKFB} = S_{KHDG} = S_{KGCF} \left(= \frac{1}{4} ...
9
votes
1answer
179 views

The smallest parallelogram that contains a convex quadrilateral

I try to find the smallest parallelogram in terms of area that contains a convex quadrilateral(A,B,C,D). I am pretty sure it must be constructed from two neighboring sides of the quadrilateral. But ...
8
votes
4answers
4k views

How to find the shaded area

How to find the shaded area crossed by semi-circle of radius 2 and quarter-circle of radius 4?
8
votes
3answers
230 views

Olympiad level | Similar Triangles

The bisector of angle $BAD$ in parallelogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. Prove that the center of the circle passing through the points $C$, $K$, ...
8
votes
2answers
273 views

Arbelos and its angle bisector

I have recently been reading about a very interesting geometry problem and have tried to solve it. I'm now in a point, in which I don't know how to move forward and would appreciate if someone could ...
7
votes
3answers
3k views

Where do you see cyclic quadrilaterals in real life?

I've just been studying cyclic quads in geometry at school and I'm thinking see seems pretty interesting, but where would I actually find these in the real world? They seem pretty useless to me...
7
votes
2answers
98 views

Given a trapezoid with base $AD$ larger than side $CD$. The bisector of $\angle D$ meets $AB$ at $K$. Prove $AK > KB$

We have a trapezoid $ABCD$ with base $AD$ larger than side $CD$. The bisector of $\angle D$ intersects side $AB$ at point $K$. Prove that $AK>KB$. All that I have tried was to make such drawing in ...
7
votes
1answer
340 views

The hunting of “missing primes”

First, I would like to introduce a peculiar way to display the prime numbers (greater than $9$) by means of the ten they belong to ($x$-axis), and their ending digit ($y$-axis). Here's an example of ...
7
votes
2answers
106 views

Maximize area of a quadrilateral given three sides

What is the maximum possible area that a quadrilateral can have, if the lengths of three of its sides are given as 3, 4 and 5, while the fourth side can have arbitrary length? (Thinking of it as three ...
7
votes
2answers
1k views

Calculating area of quadrilateral when distance of vertices from an arbitrary point is known

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 the quadrilateral. My ...
6
votes
2answers
5k views

What is the quadrilateral formed by the angle bisectors of a parallelogram?

I have drawn a few parallelograms and their angle bisectors in Geometer's Sketchpad. The quadrilateral looks to me to be a rectangle but how can I prove it ?
6
votes
0answers
112 views

Does a point quadrilateral form a rect in 3D space?

I have 4 points with x and y coordinate and would like to find out a way to check if given quadrilateral would be a rectangle in 3D space. I tried a bunch of conditions, but there was always and edge ...
5
votes
2answers
537 views

How can $4$ points in the plane be vertices of $3$ different quadrilaterals?

Four points on the plane are vertices of three different quadrilaterals. How can this happen? The problem is taken from "Kiselev's Geometry - Book I : Planimetry" At first, I thought it could be ...
5
votes
3answers
568 views

A line segment with a length of 24 makes a 90-degree angle with one of the legs of an isosceles trapezoid. What is the area of this Trapezoid?

Given that $ABCD$ is an isosceles trapezoid and that $|EB|=24$, $|EC|=26$, and m(EBC)=$90^o$. Find $A(ABCD)= ?$ From the pythagorean theorem, I can find that $|BC|=|AD|=10$. Then, I can find the area ...
5
votes
2answers
252 views

Why are parallelograms defined as quadrilaterals? What term would encompass polygons with greater than two parallel pairs?

It seems the definition of a parallelogram is locked to quadrilaterals for some reason. Is there a reason for this? Why couldn't a parallelogram (given the way the word seems rather than as a ...
5
votes
1answer
66 views

Solve for $x$ from this quadrilateral without using law of cosines.

Solve for $x$. I would be able to solve this with law of cosines (with a LOT of work!) but the students that this problem was presented are not familiar with law of cosines. Is there something I am ...
5
votes
3answers
211 views

If a quadrilateral has a pair of equal opposite sides, and a pair of equal opposite angles, then is it necessarily a parallelogram?

I’m sorry I couldn’t upload a photo, so I’ll try to explain it as best as I can. The quadrilateral has a pair of opposite and equal sides, and has a pair of opposite equal angles (85 degrees in the ...
5
votes
1answer
724 views

Proof of Ptolemy's inequality?

Can anyone prove the Ptolemy inequality, which states that for any convex quadrulateral $ABCD$, the following holds:$$\overline{AB}\cdot \overline{CD}+\overline{BC}\cdot \overline{DA} \ge \overline{AC}...
5
votes
1answer
3k views

Sides of a quadrilateral

In a triangle, with sides say $a,b,c$ we know that $a+b\geq{c}$ and $|a-b|\leq{c}$. What are the inequalities we can form given the sides of the quadrilateral say $a,b,c,d$ where these are unknown to ...
5
votes
1answer
2k views

Prove that the quadrilateral whose vertices are the midpoints of the sides of an arbitrary quadrilateral is a parallelogram

Prove that the quadrilateral PQRS, whose vertices are the midpoints of the sides of an arbitrary quadrilateral ABCD, is a parallelogram. This is an exercise in a linear algebra textbook so I would ...
5
votes
2answers
93 views

Find the angle in a quadrilateral

This is the picture, and we are aiming for the angle $x$ It's easy to see that $\angle DGA = \angle CGB = 100°$, $\angle CGD = \angle AGB = 80°$, $\angle CBG = 50°$, but now i'm missing $\angle GBA = ...
5
votes
1answer
233 views

Prove the triangle is equilateral given that a quadrilateral related to its circumcircle is a kite

Let $\triangle ABC$ be a triangle. Let $Γ$ be its circumcircle, and let $I$ be it’s incenter. Let the internal angle bisectors of $∠A,∠B,∠C$ meet $Γ$ in $A',B',C'$ respectively. Let $B'C'$ intersect $...
5
votes
2answers
386 views

Quadrilateral geometry problem, couldn't solve it.

So, I got this question a little while ago and couldn't see how to solve it. The problem follows as such: "In the following figure, G is the midpoint of CD and I is the midpoint of GE. BE:EA = 4:1 and ...
5
votes
0answers
73 views

Is this result already a known theorem in geometry?

I have been playing around with geometry and I found that: Let two perpendicular lines intersect at a point that is inside a circle. Then the area of the quadrilateral formed by the vertices made by ...
4
votes
6answers
2k views

How to find missing angles in a quadrilateral

I have a quadrilateral ABCD, with diagonals AC and BD. Given are four angles: ∠DAC = 20°, ∠CAB = 60°, ∠ABD = 50°, and ∠DBC = 30°. Those are the red angles in the above image. I need to fill in all ...
4
votes
2answers
473 views

Where does this property involving quadrilaterals come from?

$ABCD$ is a square. $|AF|=6$, $|FK|=2$, and $DE \parallel AB$. What is $|EK|=?$ My geometry book has a property for this: $$|AF|^2=|FK|\cdot|FE|$$ Can you show me where does this property come from ...
4
votes
6answers
944 views

Given distances (shortest paths) between four cities, how to show that they cannot be in the same plane?

In the example below we are given distances between four cities. The author of the book says that these distances "suffice to prove that the world is not flat". Do I understand this correctly that ...
4
votes
2answers
195 views

Side length of a quadrilateral incribed on a circle

I've been doing math for 10 years now, yet every so often I get stumped by a "basic" high school question. This is one of those times. Here's the question: Part a is easy; we apply the cosine rule ...
4
votes
2answers
157 views

A “chord” of a square, subtending a $45^\circ$ angle at a vertex, determines a triangle whose area is bisected by the square's “other” diagonal

The following Image shows the square ABCD, a point E on the side BC and the segment AF, which is a rotation of AE, at 45° (not necessarily congruent). DB is the diagonal of the square; G and H are the ...
4
votes
2answers
3k views

Ways to create a quadrilateral by joining vertices of regular polygon with no common side to polygon

How many ways are there to create a quadrilateral by joining vertices of a $n$- sided regular polygon with no common side to that polygon? It's quite easy to solve for triangles for the same question,...
4
votes
1answer
58 views

Cyclic quadrilateral and trapezoid

A circle with diameter the minor base $CD$ of a trapezium $ABCD$ intersects its diagonals $AC$ and $BD$ in, respectively, their midpoints $M$ and $N$. The lines $DM$ and $CN$ intersect in $P$ and $AC$ ...
4
votes
2answers
115 views

Are of quadrilateral: $S \leq \frac{(a+b)(c+d)}4 $

I got stuck on this problem: Given a convex quadrilateral of area $S$ and sides $a$, $b$, $c$ and $d$, prove that: $$S \leq \frac{(a+b)(c+d)}4$$ What I've done so far was to proof that $$S ...
4
votes
1answer
76 views

Show that three circles are coaxal

Let $A_1, A_2, A_3, A_4$ are collinear, $B_1, B_2, B_3, B_4$ are collinear. Such that $A_1, A_2, B_2, B_1$ lie on circle $(O_1)$, and $A_3, A_4, B_4, B_3$ lie on circle $(O_2)$. Let $MNPQ$ be the ...
4
votes
1answer
193 views

Common meeting point for 3 points to reach 4th point [closed]

Problem statement: We are 3 friends at 3 different locations $A, B, C$ and want to reach a location $D$. Each person will take a separate cab to a common meeting point $E$, and then take a single cab ...
4
votes
2answers
194 views

Testing whether the circumcenter of a cyclic quadrilateral lies inside it

For a triangle with sides $a, b, c$ (where $c$ is the biggest side) there is a simple check to see whether it's circumcenter lies inside of it: $$a^2 + b^2 < c^2$$ Is there such an inequality for ...
4
votes
1answer
37 views

Quadrilaterals that has congruent opposite sides is parallelograms

Quadrilaterals that has congruent opposite sides is parallelograms. The following is a proof. for quadrilateral ABCD, AB = CD, BC = AD, AC = AC hence ABC = CDA (SSS) mBAC = mDCA (alternate ...
4
votes
1answer
226 views

$ABCD$ is a quadrilateral and $P,Q$ are midpoints of $CD, AB.$…

$ABCD$ is a quadrilateral and $P,Q$ are midpoints of $CD, AB.$ $AP$ and $DQ$ meet at $X, BP$ and $CQ$ meet at $Y.$ Prove that $$|ADX|+|BCY|=|PXQY|$$ (here $|N|$ means area of the shape $N$) I have ...