Questions tagged [quadrilateral]
For questions about general quadrilaterals (including parallelograms, trapezoids, rhombi) and their properties.
751 questions
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Perpendicular segments from special points in a rectangle with ratio $BC = 2AB$
Consider a rectangle $ABCD$ with $BC = 2AB$. Let $L$ be the midpoint of side $AD$. From $L$, draw a perpendicular to diagonal $AC$ that intersects:
$AC$ at point $K$
$BC$ at point $F$
Let $M$ be the ...
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How many regular rhombic polyhedra exist?
Here are my definitions for regular, semi-regular, and irregular polyhedra:
A regular polyhedron is a convex, non-intersecting 3-dimensional shape made with polygon faces connected at edges and ...
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Ellipse inscribed in a convex quadrilateral
I am considering the problem of determining the ellipse that is inscribed in a given convex quadrilateral, which in addition has a certain orientation of its axes.
It is known that there is an ...
user1709783
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A formula for the diagonal of a skew quadrilateral
Let $i,j,k,m\in\mathbb R^3$. Write $\ell_{ab}=\|a-b\|$ for edge lengths, $A_{ijk}$ for the area of $\triangle ijk$, and let $\theta$ be the dihedral angle along edge $ij$ between the oriented ...
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Four lines form obtuse triangles in all triples, and Newton line doesn't intersect polar circle, then eccentricity of inscribed conics has a maximum
Let the following four distinct lines be given,
$$
\begin{aligned}
L_0&:\; 113x - 994y + 24 = 0,\\
L_1&:\; 459x - 888y + 967 = 0,\\
L_2&:\; -828x - 561y - 620 = 0,\\
L_3&:\; -182x - ...
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Quadrilateral always exists within another quadrilateral
Say we have a convex quadrilateral $ABCD$ and a set of points $S$ (no three of which are collinear) such that all $P \in S$ are in the closed set of $ABCD$, and that the points $E$, $F$, $G$, and $H$ ...
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Intuition behind reflecting $D$ in IGO $2023$ geometry problem
Problem: Let $ABCD$ be a convex quadrilateral. Let $E$ be the intersection of its diagonals.
Suppose that $CD = BC = BE$. Prove that $AD + DC ≥ AB$.
My approach: I tried to make a triangle which has a ...
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If $ABKC$ is inscribed in circle $(O)$ and $AOIK$ is a cyclic then $BC$ is the bisector of $\widehat{AIK}$
Let a quadrilateral $ABKC$ be inscribed in a circle $(O)$ and $I$ is midpoint of $BC$. Suppose $AOIK$ is a cyclic quadrilateral. I want to show that $BC$ is the bisector of $\widehat{AIK}$.
My idea :
...
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If $ABCD$ is a harmonic quadrilateral and $I$ is midpoint of $BD$ then $\widehat{BAC}=\widehat{IAD}$
Recently I've been studying about the harmonic quadrilateral and symmedians in a triangle. I want to show that : let $ABD$ be a a triangle and $I$ is midpoint of $BD$, then $ABCD$ is harmonic ...
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How to use Miquel's theorem to solve this problem?
I found this problem on a fb group and the author asked to use Miquel's theorem to solve it:
let $ABCD$ be a quadrilateral with $AB=AD$, and suppose $\angle BCD=\tfrac{1}{2}\bigl(180^\circ-\angle BAD\...
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Geometry problem from the /tttt/ board's mock Putnam exam
Let P be an interior point of the acute triangle ABC such that ∠BPC = 90°, and ∠BAP = ∠PAC. Let D be the projection of P over the line segment BC. Let M and N be the incenters of triangles ADB and ADC,...
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Geometry - Quadrilateral diagonals such that one divides the other into equal parts
Given is a quadrilateral ABCD.
Its diagonals intersect at point O.
It is given that: AO = CO
and: angle(ADC) > angle(ABC)
From the given data, what can we deduce about the relationship between DO ...
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Geometry - A variant of Langley's Adventitious Angles
I've stumbled upon a variant of Langley's Adventitious Angles problem (also known as The World's Hardest Easy Geometry Problem), see figure. Usually, this problem is solved by constructing a set of ...
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$a^2+b^2=c^2+d^2$ and $a^2+d^2−ad=b^2+c^2+bc$. Find the nearest integer value of the expression $\frac{ab+cd}{ad+bc}$
If $a,b,c,d$
are positive reals such that
• $a^2+b^2=c^2+d^2$
• $a^2+d^2−ad=b^2+c^2+bc$
Find the nearest integer value of the expression $\frac{ab+cd}{ad+bc}$.
I have seen a solution. It goes like ...
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Found simple quantities to check if a plane quadrilateral is convex, concave or self-intersecting [closed]
I have discovered two quantities that are very easily computable from the vertex coordinates of a plane quadrilateral which allow to check directly if the quadrilateral is convex, concave, or self-...
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The diagonals of convex $ABCD$ meet at $O$, and $F$ is on $AC$. Given $BO$, $DO$, $AC$, and the ratio of areas of $ABCD$ and $FBC$, can we find $AF$?
Here is the problem I'm solving. Is there even a general solution?
The diagonals of a convex quadrilateral $ABCD$ intersect at point $O$. Point $F$ is placed on diagonal $AC$. Given $BO = a$, $DO = b$...
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Prove that 5 lines are concurrent
The Newton line of a complete quadrilateral is the line joining the midpoints of the three diagonals.
There are five straight lines on a plane, any four of them determine a complete quadrilateral, ...
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Overlapping Similar Quadrilaterals
I have a question about the overlapping similar quadrilaterals. I know how to solve the problem of overlapping similar triangles using SSS, SAS, AAA, etc, but this time I am having trouble of solving ...
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Find the angle of ACD
I want to find $\angle ACD$. Given quadrilateral $ABCD$ as picture above. Let $DC=DB=AB$. Given $\angle ADB=78^\circ, \angle DAC=48^\circ, \angle CAB=30^\circ, \angle ABD=24^\circ$.
I just can only ...
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Finding the arc lengths induced by cyclic quadrilateral, given ONLY the angles of the quadrilateral
I have a quadrilateral inscribed in a circle and I want to find the arc lengths $\overset{\large\frown}{QR}$, $\overset{\large\frown}{RS}$, $\overset{\large\frown}{ST}$, and $\overset{\large\frown}{TQ}...
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Prove the geometric relation in a convex quadrilateral with midpoints of diagonals
In the figure, $ABCD$ is a convex quadrilateral and E and F are respectively the midpoints of diagonals $AC$ and $BD$.
Prove that:
$$AB^2 + BC^2 + CD^2 + DA^2 = AC^2 + BD^2 + (4*EF^2)$$
(Hint: Use ...
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properties of cyclic quadrilaterals
I have identified three theorems that seem very similar, and I would like to explore further extensions of this type of problem. Could anyone provide advice or suggest possible approaches? Or are ...
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Given the function values at vertices, how do I approximate an integral of function over a triangle/quadrilateral, consistently?
Consider following triangle PQS:
I have a function $f(\vec{r})$, which I want to integrate over this triangle. But I only know the function value at its vertices and cannot evaluate it elsewhere. A ...
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How to calculate the area of a quadrilateral given the (x,y) coordinates of its vertices
This problem is from the 2017 Gauss Contest (Grade 7).
Four vertices of a quadrilateral are located at (7,6), (−5,1), (−2,−3), and (10,2).
What is the area of the quadrilateral in square units?
I ...
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$a,b,c,e$ and $a,c,d,e$ form a concave quadrilateral then $a,b,d,e$ form a concave quadrilateral?
Let $a,b,c,d,e$ be five lines on the plane with distinct slopes,
If $a,b,c,e$ form a concave quadrilateral and $a,c,d,e$ form a concave quadrilateral, does it imply that $a,b,d,e$ form a concave ...
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