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

369 questions
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### Construct or prove existence of a certain quadrilateral

I have three questions about a quadrilateral with the following properties: It is convex. It has exactly one pair of congruent opposite sides. It has exactly one pair of congruent opposite angles. It ...
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### Value of one angle of a quadrilateral, when three sides are quadrilateral are given

In quadrilateral $ABCD$, $AB=BC=CD$. The value of $∠BAC$ is $40º$ and the value of $∠CAD$ is $30º$ what is the value of $∠ADC$?
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### The equations of two sides of parallelogram are $2x-3y+7=0$ and $4x+y=21$ and one vertex is $(-1, -3)$. Find the other vertices.

I already saw this question here, but the answers are not clear for me. Hope I could get clear solution and answer. I already got the $2^{nd}$ vertex which is $(4, 5)$. How do I get the other two ...
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### How to prove that the line joinning the midpoints of the diagonals of trapezium is parallel to the parallel sides of the trapezium?

How to prove that the line joining the midpoints of the diagonals of trapezium is parallel to the parallel sides of the trapezium? I tried to prove this by drawing BD and joining the midpoints of AD ...
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### Let ABCD be a cyclic quadrilateral…

Let ABCD be a cyclic quadrilateral. Let r and s be the lines obtained by reflecting AB through the angle bisectors of $\angle CAD$ and $\angle CBD$, respectively. Let P be the intersection of r and s ...
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### 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 ...
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### Find a the coordinate of a missing point (R) in quadrilateral [closed]

$PQRS$ is a quadrilateral with vertices $P(1,3)$ $Q(5,5)$ $R(x,1)$ $S(4,-3).$ Question: Find the coordinates of $R$ if the area of quadrilateral $PQRS$ is $42$ square unit . How to find $x$ in ...
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### 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...
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### Finding position of 3D point constrained by “parent” point constrained to circle and rotation

I have the following 3D geometry question from a camera positioning problem: (Related to this question which I got an answer to, but I was not able to extrapolate to 3D) Point $P_1$ (parent) can ...
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### Finding position of 2D point constrained by “parent” point constrained to circle and rotation

I have the following 2D geometry question from a camera positioning problem: Point $P1$ (parent) can only be on a circle about the origin with given radius $R$. Point $P2$ (child)'s position is ...
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### Is one diagonal of a convex quadrilateral always longer than at least 3 sides?

I am solving a problem which requires a proof of the statement in the title. So far, I was considering the following cases: 4 angles are at least 90°: rectangle, the statement holds. 3 angles are at ...
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### Find the measure of $DME$

In the parallelogram $ABCD$ $A$ is the small angle and $BC=2AB$. We draw the height $CE$ to $AB$.We draw a line from $C$ and $E$ to $M$. if $M$ is the midpoint of $AD$ find the measure of $DME$ in ...
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### Convex quadrilateral with interior point

Let $ABCD$ be a convex quadrilateral and $X$ is an interior point. Also let $AX\cap BD=\{E\}$, $BX\cap AC=\{F\}$, $CX\cap BD=\{G\}$ and $DX\cap AC=\{H\}$. Prove that: AF\cdot BG\cdot CH\cdot DE=...
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### A triangle in a square

The following quadrilateral is a square also there are some known angles.prove that The segments of the inner triangle are equal. My Attempt:If we name the inner point $O$ then two triangles $AOD$ ...
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### 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 ...
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### proof that diagonals of a quadrilateral are perpendicular if $AB^2+CD^2=BC^2+AD^2$

proof that diagonals of a quadrilateral are perpendicular if $AB^2+CD^2=BC^2+AD^2$. My Attempt:we know that if diagonals of a quadrilateral are perpendicular then we have $AB^2+CD^2=BC^2+AD^2$.But ...
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### In a parallelogram, do the diagonal bisect the angles that they meet?

I understand the following properties of a parallelogram: Opposite sides are parallel and equal in length. Opposite angles are equal. Adjacent angles add up to 180 degrees therefore adjacent angles ...
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### Show that if AB+CD=AD+BC then the quadrilateral $ABCD$ is tangential

Consider a convex quadrilateral $ABCD$ . Show that the quadrilateral $ABCD$ is tangential if and only:$AB+CD=AD+BC$
Problem: Consider a rhombus (Diamond) such that each of its side is the geometric mean of its diameters. I mean if length of each side is X and the diameters a and b; then $X^2$ = a.b Find the ...