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

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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 ...
• 24.4k
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 ...
• 24.4k
184 views

1 vote
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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 ...
1 vote
38 views

### 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 ...
• 13
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### 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 ...
• 139
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### 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 ...
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 ...
84 views

### 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 ...
• 24.4k
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 ...
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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 ...
• 24.4k
1 vote
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 ...
• 1,220
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### 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 ...
• 6,782
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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 ...
• 481
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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 ...
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 ...
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• 3,047
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### 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 ...
1 vote
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 ...
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} ...
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### 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 ...
1 vote
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 $MB+ MC=MA$ ...
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### 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 ...
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