Questions tagged [circles]
For questions concerning circles. A circle is the locus of points in a plane that are at a fixed distance from a fixed point.
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Does the Arctic Circle Theorem have anything to do with the geographical Arctic Circle?
Does the Arctic Circle Theorem have anything to do with the geographical Arctic Circle?
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Questions Involving Area Of Circles
I have two questions provided below about solving area for circles. They're from a previous homework I hadn't done but I wanted to try and do them now so I can prepare for my final. Any help with ...
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Formula for two overlapping circles, creating three regions with matching area
Suppose we have two circles with radius $r$.
The centers of each circle are separated by a distance $x$, such that $0 < x < r$.
This creates a Venn diagram where you see three distinct regions, ...
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Circles1 with maximum and minimum values [closed]
I have a question regarding the maximum and minimum values of $y-3x+4$ for which point $(x,y)$ moves along circle $(x-2)^2+y^2=1$.
I can calculate the question but I do not know where to locate the ...
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Geometry Challenge: Parallelogram in $\triangle$ $ABC$ with circumscribed Circle
I hope this message finds you in high spirits. I am writing to seek your expertise in solving a captivating geometry problem that I recently encountered in a competitive exam. Despite my best ...
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Surface area of sphere coming out as $\pi^2 r^2$ [duplicate]
Take a hemisphere and divide its surface area into strips like on a watermelon.
Each strip can be approximated as a triangle with the long two sides = $\pi \frac r2$ (quarter of circumference) and if ...
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Can We Prove These Removed Areas Are an Arc?
[I've been delving into math during my free time and came across an intriguing problem involving the perimeter of a shaded region. This particular challenge is part of a module that focuses on arc ...
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Circle touching a pair of lines [closed]
Let a circle S whose centre is (h,k), k>0 touches a pair of line 4xy - 3y² + 12 - h = 0 at A and B
If 'r' is the radius of the circle S then the value of (r+k) is equal to
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How close is this to squaring a circle? [closed]
How close to classically defined "squaring a circle" is this?
SQUARING A CIRCLE
To begin, take a circle of a known size (A), say 360 mm, and draw a line (B) from the top of the circle, ...
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Radius of circle given horizontal distances of two points and angle between them
Problem
I'm not certain this is possible to solve (and indeed, if the answer is "don't be silly, that's not possible" then at the very least it'll save me some time!).
I am attempting to ...
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Simple bisection geometry
Let $\triangle ABC$ have incenter $D$ and let the incircle intersect sides $BC,AB,AC$ at $E,F,G$ respectively. Extend $AB$ and $AC$ to meet the circumcircle of $\triangle ADE$ at $K$ and $I$ ...
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Prove that a convex quadrilateral $ABCD$ with certain properties is a cyclic quadrilateral [closed]
$ABCD$ is a convex quadrilateral such that $\angle ABD = \angle DBC$, $AD=CD$ and $AB \neq BC$. Prove that $ABCD$ is cyclic.
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Show that, given four points, we can always draw two intersecting circles whose diameter endpoints are the four points.
Show that, given four coplanar points, we can always draw two intersecting circles coplanar with the points, such that two of the given points are diameter endpoints of one circle, and the other two ...
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What's wrong with this picture? Impossible circles.
The planar diagram shows points $A,B,C,D$ (they don't have to be the vertices of any particular kind of quadrilateral) and four circles: $\text{C}_{AB}$ with diameter $AB$, $\text{C}_{BC}$ with ...
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Proving that the cyclic quadrilateral $ABCD$ is a parallelogram, and thus it is a rectangle.
Let us say that we have a circle and a cyclic quadrilateral $ABCD$ where the segment $BD$ passes through the center of the circle, thus being the diameter. Also, $\angle A$ and $\angle C$ are both ...
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Triangulating the surface of intersection of a sphere with edges of a cube
I want to triangulate the surface of intersection of a sphere with edges of a cube(image below). When number of intersection points > 3, the points are non-coplanar and the triangulations are non-...
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Angular width of a circle inside a circle
I would like to find the angular width of a circle inside another circle. That is, given a circle $A$ and circle $B,$ where $B$ is entirely inside $A,$ what are $0 \le \alpha \le \beta \le 2\pi$ that ...
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Geometry: Show that 2 lines are perpendicular [duplicate]
In triangle $ABC$, where $B'$ and $C'$ are the feet of the interior angle bisectors of angles $B$ and $C$, $I_a$ is the intersection point of the exterior angle bisectors, and $O$ is the circumcenter ...
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How to prove a circle in the complex plane tangent to the real axis at the origin is invariant by the function $\frac{\omega}{1-\omega}$? [closed]
Let $C$ be a circle in the complex plane $\mathbb{C}$ whose centre is $(0,ir)$ (where $r$ is a non-zero positive real number) and is tangent to the real axis at the origin $(0,0)$ (the radius of the ...
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Help understanding possibly incomplete proof in coordinate geometry of one-parameter family of circles
If $S_1=0$ is a circle and $S_2 = 0$ is another circle and they intersect at two points- A and B, then the family of circles passing through A,B is represented as
$$S_1 + k*S_2=0$$
The proof in my ...
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How do I determine the plastic section modulus Z of an arc? [closed]
I am trying to find a simplified formula for determining the plastic modulus, Z, of an arc of a circle.
In my case, I have a circle that has a diameter of 20" and I am removing an arc length of 6&...
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Conjecture: Given any $2n+1$ points, we can always draw $n$ non-intersecting circles whose diameter endpoints are $2n$ of those points.
Is the following conjecture true or false:
Given any $2n+1$ coplanar points, we can always draw $n$ non-intersecting circles coplanar with the points, whose diameter endpoints are $2n$ of those ...
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What is the largest disk that will be completely covered by randomly placed disks of areas $1,\frac12,\frac13,\dots$ with probability $1$?
On a "bottom" disk of area $A$, we place "top" disks of areas $1,\frac12,\frac13,\cdots$ such that the centre of each top disk is an independent uniformly random point on the ...
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Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points.
Is the following conjecture true or false:
Given any five coplanar points, we can always draw at least one pair of non-intersecting circles coplanar with the points, such that two of the given points ...
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Prove some points are concyclic
question
We consider the triangle $ABC$ with $AB$, $AC$ and in which $(AX$, $(AY$, two half lines inside the $\angle BAC$, isogonal (ie: $\angle BAX= \angle CAY$). We consider the points $E$ and $F$ ...
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How to express $y$ from $x^2+y^2>r$?
Consider an inequality in form of circle equation (with zero coordinates):
$$y^2 + x^2 > 1$$
Expressing $y$ step-by-step (maybe I wrong somewhere?):
$$y^2 > 1 - x^2 (1)$$
$$y>\sqrt{(1-x^2)}\...
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Determine the circle that is tangent to three given ellipses
Given three ellipses in the plane specified as follows
$(r - C_1)^T Q_1 (r - C_1) = 1$,
$(r - C_2)^T Q_2 (r - C_2) = 1$
$(r - C_3)^T Q_3 (r - C_3) = 1$
I want to find the circle that is externally ...
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Prove or disprove: There is a way to choose independent random chords in a circle so that their intersections are uniformly distributed in the circle.
Prove or disprove:
There is a way to choose independent random chords in a circle so that
their intersection points (given that they exist) are uniformly distributed in the circle.
One common way to ...
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Why is the probability that two random chords intersect $2/3$?
In a circle, a random chord is drawn by connecting a uniformly random point on the circle, with a uniformly random point on the opposite semicircle. (So the chord's length is at least $\sqrt2$ times ...
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Why is there no $\pi$ in an ellipse?
Allow me to clarify ...
With a circle of circumference $c$ and diameter $d$, $\pi$ makes an appearance as $\frac{c}{d}$ even though the equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$, doesn't ...
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Find the center and radius of a circle tangent to three given semi-circles
Three semi-circles have their centers as follows: $(0,0), (2, 0), (-4, 0)$ with radii of $6, 4, 2$, respectively. Determine the radius and center coordinates of a circle that is tangent to all three ...
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How do I calculate the intersection points between inverse shared tangent lines, a circle, and a bounding box?
Context
I am attempting to draw a polygon with 4 or 5 corners using a software library that draws shapes by taking in an array of corners in the form of $(X,Y)$. I am somewhat woefully under skilled ...
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Intuition is silent: Find the probability that the smallest circle enclosing $n$ random points on a disk lies completely on the disk, as $n\to\infty$.
On a disk, choose $n$ uniformly random points. Then draw the smallest circle enclosing those points. (Here are some algorithms for doing so.)
The circle may or may not lie completely on the disk. For ...
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Euclidean Geometry - Construction of a Circle Tangent to Given Circle, Line, and Point on Line
This is the original phrasing of the question:
"Describe a circle to touch a given circle, and also to touch a given
straight line at a given point."
A School Geometry H.S. Hall and F.H. ...
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How to find a circle’s tangents at points on its circumference?
I have a circle in 3-space (the coordinates of its center, its radius, and a vector normal to its plane). I also have several points on that circle. Now I want to find the unit tangents to the circle ...
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Analyzing a Complex Function: Does f(z) Preserve Circles and Invariant Sets?
Let $f: \mathbb{C} \to \mathbb{C} $ be given by $$f(z) = \frac{2iz}{z + i}$$,
$C$ be an arc of a circle passing through $0$ and $i$, and $$ A = \left\{ z \in \mathbb{C} : \text{Re}(z) > 0 \land \...
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Circumcentre of Triangle Formed by Tangents of Intersecting Circles at the point of intersection
I faced the above question in my textbook-
Two circles with centres O and Q intersect at two points, one being A. Their tangents at A meet the circles again at B and C respectively. A point P is ...
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how do i find the equations of L1 and L2 [closed]
L1 and L2 are perpendicular. the equation of the circle is given as $x^2+6x+y^2-2y=7$. line L1 cuts the circle at $P$. L2 cuts the circle at $Q$. I need to find the equations of the lines L1 and L2. I ...
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Billiard Shot Angles for Circular Table: Return to Starting Point
A circular billiard table is given with a cue ball at the circumference. It is shot at an angle of θ to the line from the ball to the center of the table. For what angles θ will the ball return to ...
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Is there a name for a region of a circle comprising an inscribed angle subtending a chord, and a corresponding circular segment? ("ice-cream cone"?)
Is there a name for the particular part of a circle shown in red? I know the white sections are segments. If the vertex was at the center of the circle it would be a sector.
I don't suppose that any ...
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What is the maximum number of squares that can fit in a circle?
The conditions for this are
No overlapping
The answer can be in decimals (so the number of squares doesn't have to be whole)
The radius of the circle and the side length of the square is same
My ...
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In acute $\Delta ABC$, $O$ is circumcentre. $M,N$ on $AB,AC$, s.t. $O$ lies on $MN$. $D,E,F$ are midpoints of $MN,BN,CM$. Prove O,D,E,F are concyclic.
In acute $\Delta ABC$, $O$ is the circumcentre. $M,N$ lie on $AB,AC$ respectively, s.t. $O$ lies on $MN$. $D,E,F$ are midpoints of $MN,BN,CM$ respectively. Prove that $O,D,E,F$ are concyclic.
I was ...
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Is this a new point on the nine-point-circle of a triangle?
I was trying to get a feel for how to solve another question about the largest triangle that can fit in a unit square, by constructing the smallest enclosing square of a triangle in Geogebra. While ...
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Find center of externally tangent circle
I've been struggling to find a way to resolve the following problem:
Let $C_1$ a circle of center $V$ and of radius $r_1$.
Let $A$ and $B$ two points outside of $C_1$, and $L$ a line passing by them.
...
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Parametric equations for a point traversing the circumference of a laterally accelerating circle (not rolling) with a max speed for the point.
Variables:
e -> point on a circle
r -> radius of said circle (constant)
x(t), y(t) -> parametric equations describing the position of point e
v -> velocity of the center of the circle
c -&...
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Locus of circle using complex numbers
I was going through Locus of circle using complex numbers and got stuck on one particular manipulation.
We know that $|z-z_1|=r$ represents a circle whose centre is $z_1$ and radius r. Now in the book ...
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Finding the diameter of a circle projected on a plane
I am an engineer working on a machine designed to measure the diameters of cylindrical bores. To provide a clear and engaging context for my question, I'll describe the physical scenario in detail.
...
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Maximum distance (gap) possible b/w two athletes when they are running infinitely in a circle of fixed radius
There are two athletes (say A & B) - starting from same fixed point-running with speed X m/s and Y m/s around a circle of radius R in anti-clockwise direction - infinitely. How can I find the ...
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Is it possible to explain geometrically why $\tan^2\alpha_T + \tan^2\beta_T =$const.?
Recently I have found an old journal of Serbian Mathematical Society, and there is one neat award task for high school students.
A circle $k$ is inscribed in a square $ABCD$. For an arbitrary point $...
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Differential equation of orthogonal curves
Let us suppose we want to find the curve which is perpendicular to the curve $y=3x$. From the concept of straight lines,we know that the slope of that curve will be $-\frac{1}{3}$. Whose equation ...
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