For questions conserning circles. A circle is a curve composed of points in a plane that are at a fixed distance from a fixed point.

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A generalization of the first Droz-Frany circle

I am looking for a proof of the following problem: Let $ABC$ be a triangle with circumcenter $O$, and the medial triangle $M_aM_bM_c$. Let $O_a, O_b, O_c$ be three points on three lines $OA, OB, ...
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7answers
231 views

Area of a circle $\pi r^2$

So, today I learned that the area of a circle is $\pi r^2$. So, I thought that since $r$ is $1$ dimensional, $r^2$ will be $2$ dimensional. In this case, a square, as you only multiply $2$ dimensions ...
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4answers
75 views

How to find terminal point coordinates on a unit circle?

Hey everyone I am working on a homework assignment which covers unit circles. However I am really confused and having a lot of trouble locating terminal point coordinates. Everything I have read ...
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2answers
59 views

Eccentric circles

I have an equation to calculate the distance to the outside of a circle from an eccentric point within the circle. $$x = E\cos(a) + 0.5\sqrt{(D^2) - 4*(E^2)\sin(a)^2}$$ Where: $E$ = eccentricity, ...
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1answer
34 views

What triangles can be cut into three triangles with equal radii of the circumscribed circles around these triangles?

What triangles can be cut into three triangles with equal radii of the circumscribed circles around these triangles? My work so far: Case 1) let $ABC -$ an acute-angled triangle. Then radii of the ...
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4answers
86 views

Finding $x, y$ coordinates in a circle

I have a laser that will be measuring distances in a circular tank to identify unique locations. The laser will take north $(y)$ and west $(x)$ measurements and then it can be rotated if necessary to ...
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1answer
66 views

Calculate distance between two points on concentric circles

I am trying to find the shortest distance between two concentric circles. I already know the angle between the two points and radii of the circles, but I am not sure how to calculate the distance ...
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2answers
75 views

Geometry - Tangent circles

Let chords AC and BD of a circle ω intersect at P. A smaller circle ω1 is tangent to ω at T and to segments AP and DP at E and F respectively. (a) Prove that ray T E bisects arc ABC of ω. (b) Let ...
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1answer
153 views

total no. of different complete line sequences that can be drawn using n points in a circle? [duplicate]

we are given n points, 1 to n which are placed on a circle. we need to draw lines b/w the points. one point can only be a part of one line. we need to draw lines in such a way that no two lines ...
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1answer
57 views

Challenging Circle Theorem

In the given figure $PQRS$ is a cyclic quadrilateral. $PQ$ and $SR$ are produced up to center $O$ of the circle. $OT$ and $OR$ are the radii of the circle. $QR$ and $PS$ are produced upto the point ...
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2answers
51 views

Best Fitting Pipe in parabolic trench

A work crew is digging a pipeline. The cross section of the trench is in the shape of the parabola $y = x^2$. The pipe has a circular cross section. If the pipe is too large, then the pipe will not ...
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4answers
56 views

Coordinate Geometry: Are there enough information to find out the coordinates?

Question: Given the circle $x^2+y^2=25$ is inscribed in triangle $\triangle ABC$, where vertex $B$ lies on the first quadrant. Slope of $AB$ is $\sqrt 3$ and has a positive y-coordinate, and ...
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0answers
50 views

The arc length of a circle section if radius is changing?

I would like to find the angle subtended by an arc of a circle with a changing radius. The main issue is that the radius is changing by a non-linear factor as shown below: The integral on the left ...
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2answers
98 views

A chain of six circles associated with a cyclic hexagon

I found the problem some months ago. But I never have been a proof. So I am looking for a proof. The problem as following: Let $ABCDEF$ be a cyclic hexagon. Let $(C_{AD})$, $(C_{BE})$, $(C_{CF})$ ...
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2answers
69 views

How to place the biggest number of cans in this box

A rectangular box has dimensions of 108 cm x 144 cm for its bottom. I want to place the biggest number of cans of 12 cm of diameter in it. How can I place this biggest number? It couldn't be as ...
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2answers
29 views

A circle of finite radius with points $(-2,-2),(1,4),$ and $(k,2006)$ can exist for

A circle of finite radius with points $(-2,-2),(1,4),$ and $(k,2006)$ can exist for $(A)$ no value of $k$$(B)$exactly one value of $k$ $(C)$exactly two values of $k$ $(D)$infinite values of $k$ Let ...
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1answer
43 views

How to solve angle from area of circular segment formula?

I know the radius $R$ of the circle and the area $A$ of the segment. How can I solve for central angle $\alpha^{\circ}$ in this (or some other) equation: $$A=\frac{R^{2}}{2} \left( \frac{\alpha ...
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1answer
90 views

Relationship between incenter and circumcenter

Let ABC be an acute triangle with circumcenter O and incenter I. Points E, M lie on AC and F, N on AB so that BE ⊥ AC, CF ⊥ AB, ∠ABM = ∠CBM and ∠ACN = ∠BCN. Prove that I lies on EF if and only if O ...
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2answers
33 views

How to determine the angle of intersection? (2 circles)

Here's an example of what I have: The radius of both circles is 50. Each circle is moved in by 10, so the distance between the two center points is 80. As you can see one of the circles end at 90 ...
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2answers
44 views

Finding sides of triangle

Given : $$\triangle ABC$$ $$M \in AB,N \in BC ,P \in AC$$ are the points at which the incircle crosses the triangle $$MN=3\sqrt{10}$$ $$NP=2\sqrt{20}$$ $$PM=10$$ I have to find the sides of the ...
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2answers
69 views

Modelling the difference between intersections of two lines on the circumference of a circle

I have a line which is divided into small segments. In the following diagram we have the first segment defined by two points $P_1$ and $P_2$. However, imagine the line having other segments evenly ...
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3answers
64 views

Problem Involving Circle Geometry [closed]

A circle passes through the following points: $(0,0)$ $(1,3)$ $(3,0)$ and $(2,3)$ Find the centre and radius of the circle and explain why. Thanks for any help. So far I have drawn a graph, the ...
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1answer
19 views

Circleline points

How can I show that $\forall$ distinct $z_1,z_2,z_3,z_4 \in \mathbb{C}$, $\frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)} \in \mathbb{R} \iff$ the points lie on a circle? I don't really have a clue of ...
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1answer
25 views

X is any point on AB and the median AD of triangle ABC meets XC at Y.Prove that XY/YC = AX/XB

X is any point on AB and the median AD of triangle ABC meets XC at Y.Prove that XY/YC = AX/XB.
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2answers
24 views

Complex inversion map

How do I show that the map $f: \mathbb{C}\setminus \{0\} \to \mathbb{C}\setminus \{0\} $. $f(z): z \mapsto \frac{1}{z} $ maps circles to either a circle or a line?
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0answers
28 views

Prove the following represents a circle

I'm trying to prove the following represents a circle but I can't see how to reduce it down to the form $|z-z_0|=c$: Let $r \in \mathbb{R}, r \not= 0, c \in \mathbb{R}, k \in \mathbb{C} : ...
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1answer
15 views

determine the centre points of the circle

Given: Circle with centre $M (-5; 5)$ The equation is $(x+5)^2 + (y-5)^2 = 50$ Suppose this figure is translated $6$ units to the right and $3$ units down. What is the new centre of the circle? ...
2
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1answer
24 views

triangle park problem [closed]

We have a park that is triangle. We don't know the shape of the triangle and it can have any triangle shape and lengths. Where should we place a lamp to have light everywhere in the park? my english ...
3
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2answers
34 views

Equation of a tangent line for circles

When calculating a tangent to a circle, is the method the same as tangent to a curve? Problem: A circle has a radius of $2$ and is centered at the origin. Find the the equation of the tangent line to ...
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2answers
23 views

Geometric Sum of the circumference of the layers of a cake

The diameter of each successive layer of a wedding cake is 2/3 the previous layer. If the diameter of the first layer of a 5 layer cake is 15 inches, find the sum of the circumferences of all the ...
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1answer
21 views

How to plot concentric circles (or other patterns?) on a grid of pixels, ensuring every pixel is occupied

A hobbyist programmer asks... Let's say a "pixelMap" is an array of x,y coordinates in a square region at which to render each color that's read from a separate array (in order from start to finish) ...
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3answers
46 views

Finding End point of an Arc in Cartesian Coordinates while radius, arc length and one end of Arc is given?

The problem Picture I want to find the position of a robot using single tire model while rotating. I am assuming robot is moving along a circle. I know its radius, length or arc and starting point of ...
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3answers
64 views

How can I show that $AX=AY$?

In the diagram below, the two circles have equal radii. However after much bashing around with angles, I was not able to show that $AX=AY$. My only idea so far is to try to instead prove that ...
2
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2answers
72 views

How to find the area of a semicircle inside of another semicircle?

I am stumped on how to solve this problem. I've tried to look up the problem, but to no avail. What I need to know is how can I find the area of a semicircle cut off by the curve of another ...
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1answer
32 views

4 circles one thread circling them [closed]

Can somebody tell me how to measure the length of a thread which is wrapped around 4 circles the radius of each is 1m. The four circles are touching but their ...
2
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2answers
80 views

How do you calculate the area of the intersection between a rectangle and a doughnut?

I'm dealing with an engineering problem, involving concentric pipes, with air flowing through the outer pipe (doughnut). I need a cross-beam to support the inner pipe, so I need to calculate how much ...
4
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6answers
210 views

How can never ending decimal numbers represent finite lengths? e.g. pi(π), $\sqrt{2}$

Recently, I was in a discussion with a colleague that, whether the πd really can represent the accurate perimeter of a circle or not. To clarify that doubt, I came ...
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1answer
23 views

How to find the intersection coordinate with circle and line equation?

http://i.stack.imgur.com/krUqK.png Example we have one line and one circle above, and its equation are: Circle: $(x-a)^2 + (y-b)^2 = r^2$ Line: $y = m(x-x_1) + y_1$ How do we find the ...
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2answers
55 views

Find the circumference of a circle, given a chord and the length of the rest of the circle [closed]

If you know the measure of a chord in a circle as well as the length of the circumference minus the arc can you find the circumference of the circle? I can make a mockup image if it would help!
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3answers
39 views

Is this an equation of a circle?

Wanted to know why the following equation doesn't represent a circle: $2x^2 + 2y^2 − 6x + 4y + 7 = 0$ I know that $(\frac{-a}{2})^2 + (\frac{-b}{2})^2 - c \geq 0$ And it is, but the exercise says ...
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4answers
68 views

Properties of Equilateral Triangles in Circles

If there is an equilateral triangle in a circle, would the midpoint of any of the 3 sides be half the radius? e.g if the radius was 6 and at the midpoint of the triangle (call it B) would center to ...
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1answer
54 views

What is the equation of circle with radius $\sqrt{2}$, tangent to the line $x+y=3$, and having its center on the line $y=4x$? [closed]

What is the equation of circle with radius $\sqrt{2}$, tangent to the line $x+y=3$, and having its center on the line $y=4x$? Can someone help me please?
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1answer
29 views

Circles and mid bisector

Let the point $M -$ bisector middle $AD$ acute triangle $ABC$. A circle $\omega_1$ with a diameter of $AC$ intersects the segment $BM$ at point $E$, and a circle $\omega_2$ with a diameter of $AB$ ...
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1answer
32 views

minimum radius of circle tangent to 3 sides of quad

I have a convex quadrilateral defined (in counterclockwise order) by the points $$p_0=(x_0,y_0)\\p_1=(x_1,y_1)\\p_2=(x_2,y_2)\\p_3=(x_3,y_3)$$ I want to found the minimum radius between all the radius ...
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1answer
107 views

Shortest distance between two circles

What is the shortest distance, in units, between the circles $(x - 9)^2 + (y - 5)^2 = 6.25$ and $(x + 6)^2 + (y + 3)^2 = 49$? Express your answer as a decimal to the nearest tenth. So I know that ...
0
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1answer
74 views

Value Of $\pi$ obtained using limits!

What i thought was simple, a circle can be formed by increasing the number of sides of regular polygon( like pentagons, hexagons, etc ) up to infinity by keeping the distance between the center and ...
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2answers
24 views

$r<a, (x+a)^2 +y^2=r^2, (x-a)^2 +y^2=r^2$ four tangent lines

Find the equation of the four tangent lines which are tangent to both circles, $(x+a)^2 +y^2=r^2, (x-a)^2 +y^2=r^2$ Do not give it in the form that involves trigonometric ratios. What are the four ...
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1answer
51 views

How to rotate a line segment around one of the end points?

I am given x1, y1, x2, y2 and θ. How can I find x3 and y3? By the way, there can be another line segment on the other side of AB (as if the line was rotated counter-clockwise). How to find that ...
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1answer
21 views

Calculating radius of circles which are a product of Tangent Intersections using a Regular Polygon

Introduction Lets have a regular polygon of $n$ sides inscribed in a circle of radius $H$, then construct tangents between the circle and each point of the polygon and draw new circle(s) trough the ...
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2answers
31 views

If you colored every point of a circle 1 of 2 colors, is there always 2 same-colored points of distance $R$ apart?

If every point on a circle of radius $R$ in $\mathbb{R}^{2}$ were colored one of two colors, is there necessarily two points that are of the same color and of distance $R$ apart? what about $>2$ ...