Questions on the circle, a curve composed of points in a plane that are at a fixed distance from a fixed point.

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1
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2answers
46 views

Pidgeonhole principle - I'd like an explanation for this answer

A friend of mine showed me how to solve this question: suppose there are 5 black dots drawn on a blue sphere. show that there is a closed hemisphere such that 4 of the black points are in it. his ...
1
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1answer
53 views

M fibers over the circle then construct a symplectic form

I'm trying to prove that if a 3-manifold $M$ fibers over the circle, then $M\times S^1$ admits a symplectic structure. I know that it is an standard result. Probably it is very easy, but I can't see ...
0
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1answer
125 views

Circle containing three points, maybe all collinear

We all know that a circle is exactly defined by three distinct non-collinear points. But I need a way to solve the following problem (all in 2D): Given three points, calculate a circle with all three ...
2
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0answers
158 views

Can you explain the solution of this geometric problem

A year ago IBM research posted an interesting geometrical problem: A gardener plants a tree on every integer lattice point, except the origin, inside a circle with a radius of $9801$. The trees ...
3
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1answer
646 views

Solving circle's radius only knowing angle & lengths of external triangle OR solving for sides of a triangle partial side lengths

Is this possible? Given that I know the length of Y and Z and the angle of X can I figure out the radius A? If I can't without more information, I can produce another set of data X Y Z at a ...
12
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10answers
959 views

How is the value of $\pi$ ( Pi ) actually calculated?

When I was a child I was taught $\pi$ (Circumference/Diameter) is an irrational number and can be approximated to $22/7$ but $= 3.(142857)(\ldots)$. But where does this value comes from? In ...
1
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2answers
200 views

Why is the circle of convergence for complex power series a circle (and not e.g. a square)?

Power-Series have an "circle of convergence". With real numbers this is an interval. Expanding this to complex numbers this becomes a circle. There are lots of book stating this, but I did not find ...
2
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2answers
250 views

Prove that if two circles touch one another, then these chords, drawn from the tangency point, are proportional.

Prove that if two circles touch one another, then chords of the internal circle, drawn from the tangency point, are proportional to the chords of the outer circle that you get when you extend the ...
2
votes
2answers
61 views

On congruent chords

Let $C_i=C(A_i,r_i)$ two secant circles intersecting each other at $R,S$, with $r_1\neq r_2$. Let $M$ be the median point of $A_1A_2$. Let $t\perp RM$ at $R$, intersecting the circles at $X,Y$. I'd ...
0
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1answer
33 views

probability calculation for position measurement being inside a circle

Consider a position measurement that is prone to a random error in any direction. This would mean that the position would be in a circle where the probability curve taken across the diameter would ...
0
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1answer
440 views

Intersection of a parametric curve and a circle

Given a curve defined by a parametric equation $x(t)$ and $y(t)$, how might one calculate the point of intersection with a circle? The derivatives $x'(t)$ and $y'(t)$ are also available if they prove ...
6
votes
3answers
369 views

Proofs without words of some well-known historical values of $\pi$?

Two of the earliest known documented approximations of the value of $\pi$ are $\pi_B=\frac{25}{8}=3.125$ and $\pi_E=\left(\frac{16}{9}\right)^2$, from Babylonian and Egyptian sources respectively. ...
1
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1answer
40 views

stuck on a Cartesian question

we have a circle $(x-1)^2+(y-2)^2=9$ Point $P=(5,2)$ lies outside the circle. Solve the equation of the line which passes through $P$ and intersects the circle at two points whose mutual distance is ...
3
votes
5answers
407 views

How do I find/predict the center of a circle while only seeing the outer edge?

Question What formula would allow me to predict the center of this circle? In addition, what attributes of this image must be detected in order to predict the center? I figured understanding the ...
2
votes
5answers
4k views

Finding the shortest distance between a point and a circle

The question is "Find the shortest distance from the origin of the graph of the circle $x^2-14x+y^2-18y+81=0$ ". I found the circle in the following form: $(x-7)^2+(y-9)^2=7^2$ Then I found the line ...
0
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0answers
88 views

(PA)^2 + (PB)^2 +(PC)^2 + (PD)^2 is equal to?

A circle is inscribed into the rhombus ABCD with one angle 60. The distance from the centre of the circle to the narest vertex is equal to 1. If P is any point of the circle ,then $$(PA)^2 + (PB)^2 ...
2
votes
5answers
1k views

Can we prove that circle has at most two colinear points?

Ok so I stumbled across a problem(I found a solution just to be clear) and it got me thinking.The problem is a classic,it was challenge to prove that line can intersect circle in at most 2 points. So ...
0
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1answer
59 views

Fixed points through a general circle.

The circle $C: x^2 + y^2 + kx + (1+k)y - (k+1)=0$ passes through two fixed points for every real number $k$. Find $(i)$ co-ordinates of these two points and $(ii)$ the minimum value of the radius.
10
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1answer
202 views

An Unexpected Circle…

I played around with $$z=\frac{-1+e^{it}}{\phantom{-}2+e^{it}}$$ and found that, when I draw the real against the imaginary of $z$, it pretty much looks like a circle. But neither ${\frak{R}} z ...
2
votes
1answer
74 views

Contract expression of circle segment area contingent on height

I want to determine a function for the area of the segment's height. I have made it this far, but I would like to contract the equation further - sadly, I do not know how to do this while still ...
6
votes
2answers
138 views

Regular pentagon and tangent lines

We consider a regular pentagon $A_1A_2A_3A_4A_5$ and $(C)$ is its inscribed circle. We then, taking as centres the points $Α_1$,$Α_2$,$Α_3$,$Α_4$,$Α_5$, draw the circles ...
2
votes
1answer
189 views

Optimization: A certain amount of wire to create a square and a circle, minimize area.

You have 4 feet of wire to create a square and a circle. How much wire should you spend on each shape to minimize the area. Also, why isn't the minimum area when you use all of the wire on the ...
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5answers
181 views

If we are given a circle and its equation and a point which lies on it..can we find the diametrical opposite point?

If we are given a circle and its equation and a point which lies on it.. Can we find the diametrical opposite point?
2
votes
1answer
86 views

Nine-point-circle, midpoint of triangle

ABC is the triangle and M, N are midpoints of AB and AC. Points W, X are on AB, Y, Z are on AC such that WM = MX, ZN = NY. Let T be the intersection of WY and XZ, prove that T lies on the nine point ...
1
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1answer
97 views

Intersecting great circles to find position

Is it possible to find the intersection of two great circles when knowing the following: A point $a$ on earth, A point $b$ on earth, and The bearings of $a$ and $b$ from an observer?
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1answer
181 views

How to identify if points are on the left or right side of a circle

Suppose I have a series of points that are on a 2D plane, and I know they can be fitted to some part of the circumference of a circle. How can I determine that the points lie on the left or right ...
0
votes
1answer
116 views

Geometry GRE question

This is a GRE quesrtion, and I could not find the length to save my life, please help! A circle with diameter PQ of length 10is internally tangent at P to a circle of radius 20. A sqare ABCD is ...
0
votes
1answer
80 views

How to find the n number of coordinates of circumference of circle?

Line AB has two coordinates A = (1,3) and B = (1,6). How to find 10 uniform coordinates of a circumference of circle whoose radius is AB. Edit I tried this link but didn't get it.
2
votes
1answer
58 views

Bouncing of a ball from circular boundary

Lets say a ball with xspeed: 14, yspeed: 16 hits the circular edge at xposition:626 yposition:382 like on the below picture : It needs to bounce properly, to get the right bounce and new ball ...
1
vote
2answers
942 views

What is the equation for a line tangent to a circle from a point outside the circle?

I need to know the equation for a line tangent to a circle and through a point outside the circle. I have found a number of solutions which involve specific numbers for the circles equation and the ...
2
votes
2answers
185 views

Square covered with circles

I have a square 800x800 and i need to fully cover it with the least number of circles possible, each circle has a radius of 150. QUESTIONS: - What pattern would be the best to use? Clover, diamon or ...
4
votes
2answers
618 views

Expected number of points on circle to form an acute angled triangle

This problem was asked to me in an interview. We keep on adding points on a circle uniformly until there exist three points on the circle which form an acute angled triangle. What is the expected ...
1
vote
1answer
102 views

Determine if a point is contained in the circle in 3d space

I have a problem where I need to determine if a point is contained in the area of a circle in 3d space. For my circle, I have the radius (R), the position of the center (C) and a normal vector to the ...
0
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2answers
57 views

calculating various cuts of a circle

im trying to find some sort of formula to calculate lines within a circle. I need to find the length of the various lines within the circle from which I only know the diameter. Is there some sort ...
39
votes
3answers
3k views

Cutting up a circle to make a square

We know that there is no paper-and-scissors solution to Tarski's circle-squaring problem (my six-year-old daughter told me this while eating lunch one day) but what are the closest approximations, if ...
0
votes
2answers
538 views

Find equation of the circular cross section of a unit sphere

I have a unit sphere in Cartesian coordinates: $x^2 + y^2 + z^2 = 1$ or in spherical coordinates: $x = \rho \sin(\phi) \cos(\theta)\\ y = \rho \sin(\phi) \sin(\theta)\\ z = \rho \cos(\phi)$ I ...
1
vote
0answers
185 views

Problems with Circles and Lines on a Cartesian Plane

(a) Find the equations of the two circles each of which touches both coordinate axes and passes through the point $(9,2)$. (b) Find the coordinates of the second point of intersection of the two ...
1
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0answers
66 views

Probability of a triangle in a circle [duplicate]

I'm confused on my calculations on analytic geometry with probability. Things I learned on these were messed up since I was a newbie on these subjects. Here's my problem: Three points are chosen ...
1
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1answer
194 views

Furthest point on regular polygon given arbitrary direction

In a circle of radius $r$ centered at $c$, if I want to know the point on the circle that is furthest in a direction specified by a vector $d$ I use the formula $c+(r/||d||)d$. Is there a similar ...
6
votes
2answers
789 views

How exactly do you measure circumference or diameter?

I am absolutely confused about trying to calculate circumference. And I do not mean using the math formula, I mean back in old days when people had very primitive tools, and had to make the ...
0
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1answer
232 views

Calculate origin of pie slices

I need to draw a pie graph with slices similar to the following... The required information I need to draw a single slice is the X & Y coordinate of the origin of the slice and the angle of the ...
0
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1answer
50 views

Constant-Width Curves and Circles

I recently learnt that a constant-width curve is a curve with continuous constant width, therefore, I believe a circle is a constant-width curve. However, I am sure there is a property that a circle ...
1
vote
1answer
124 views

Finding the radius of a circle

Given a point A that outside a circle so that $AT$ is tangent to the circle in point $T$ And $AC$ is a secant to that circle in points $B,C$. From points $B,C$ we build heights to $AT$ ...
0
votes
1answer
57 views

Help finding the arc length?

What is the arc length if $Θ = 6\pi/5$(sorry, dont know how to format that) and the radius is $2$ cm? length of arc= $n/360= 2\pi(r)$ is it $2/360=2\pi(r) 2\pi(180)$?
1
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1answer
1k views

Inscribed Angle Theorem to prove: “An angle inscribed in a semicircle is a right angle.”

I came across a question in my HW book: Prove that an angle inscribed in a semicircle is a right angle. My proof was relatively simple: Proof: As the measure of an inscribed angle is equal to ...
0
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1answer
104 views

Proving the diameter is two times the radius

I am stuck on the following question: Prove that each diameter is twice as long as each radius. I drew a circle, with center O and diameter AB. Is there a theorem that could help me say that ...
3
votes
4answers
32k views

How to find the equation of a line tangent a circle and a given point outside of the circle

I am given the equation of a circle: $(x + 2)^2 + (y + 7)^2 = 25$. The radius is $5$. Center of the circle: $(-2, -7)$. Two lines tangent to this circle pass through point $(4, -3)$, which is outside ...
5
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1answer
1k views

Is *njwildberger* wrong about area and circumference of a circle?

In this video, njwildberger says that the area and circumference of a circle are proof-less theorems. But I heard that we can derive both the area and circumference of a circle using calculus? So are ...
0
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2answers
362 views

Diameters and Circles

I have a question (given by a teacher) that looks really easy but then when I thought about it, couldn't find a way to find the answer. It is a proof question relating to diameters: Prove that any ...
23
votes
5answers
1k views

Did Euclid prove that $\pi$ is constant?

Pi is defined the ratio of the circumference of a circle to its diameter, but of course different circles have different circumferences and diameters, so in order for it to be well-defined we need to ...