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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Prove that the circumference of a circle is $25\pi$ [on hold]

A regular hexagon inscribed in a circle has an area of $$54*3^\frac{1}{3} \text{sq.in}$$ Prove that the circumference of a circle is $$25\pi$$
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+100

Locus and Circles _ Integer Answer

Let T be the line passing through the points P(–2, 7) and Q(2, –5). Let $F_{1}$ be the set of all pairs of circles $(S_{1}$, $S_{2}$) such that T is tangent to $S_{1}$ at P and tangent to $S_{2}$ at Q,...
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2answers
46 views

If two circles meet at a unique point, then the line through their centers passes through that point

Let two circles meet at a unique point: M. Prove (or disprove) that the line joining their centers passes through M. My attempt: $T_1$ is tangent for $C_1$ and will meet $C_1$ at a single point $M$. ...
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1answer
291 views

Better method to solve a geometric problem.

This question is Q.13 of International Mathematical Olympiad Preliminary Selection Contest - Hong Kong 2019. $A$, $B$, $C$ are three points on a circle while $P$ and $Q$ are two points on $AB$. The ...
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3answers
111 views

Locus of a point where two circles having a common tangent meets

Let T be the line passing through the points P(-2,7) and Q(2,-5). Let $F_1$ be the set of all pairs of circles $(S_1,S_2)$, such that T is tangent to $S_1$ at P and tangent to $S_2$ at Q, and also ...
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1answer
112 views

Olympiad level | Similar Triangles

The bisector of angle $BAD$ in parallelogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. Prove that the center of the circle passing through the points $C$, $K$, ...
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2answers
27 views

Problem about radius and chord of a circle. [on hold]

The radius of a circle is 14. How far is a chord of length 12 away from the center of the circle?
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1answer
34 views

Calculate the outside (border) of intersecting circles

I am developing a real-time strategy game where you are able to move a number of your units around a map. When a unit is selected, I draw a circle around that unit to represent that unit's vision ...
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3answers
185 views

Find the value of the angle $X$ in the given figure [closed]

In this picture, the curve in the inside of the big outer triangle is actually its incircle. The edges of the triangle inside the incircle are the intersections of the incircle with the outer triangle....
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6answers
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The common tangents to the circles $x^2+y^2+2x=0$ and $x^2+y^2-6x=0$ form an equilateral triangle

Problem : Show that the common tangents to circles $x^2+y^2+2x=0$ and $x^2+y^2-6x=0$ form an equilateral triangle. Solution : Let $C_1 : x^2+y^2+2x=0$ here centre of the circle is $(-1,0) $ and ...
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1answer
19 views

Parametric equation of circle arc

I need to have a parametric formula that given a start point (Lat, Long), end point (Lat, Long) and center and a parameter t gives me a point in the arc between those 2 points. I need to calculate all ...
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1answer
43 views

A relation of the lengths of line segments in a circumscribed star pentagon.

A few days ago I posted a question and timon92 gave a beautiful solution. Using timon92's method, one can easily show that $ER\cdot BS\cdot CD=DE\cdot RS\cdot BC$ (as with the notations in the figure ...
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1answer
33 views

How can I convert $r = 2\cos{\theta}+2\sqrt{3}\sin{\theta}$ to cartesian coordinates? [on hold]

I'm struggling to figure out this circle equation in polar coordinates: $r = 2\cos{\theta}+2\sqrt{3}\sin{\theta}$ and converting it to cartesian form. How can I convert this to cartesian? How can ...
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1answer
30 views

How many circles can fit on the perimeter of $N$-gon

Given a regular $N$-gon with perimeter $P$ and circle with radius $R$. How many of these circles can fit on the perimeter of the $N$-gon without overlaping? $\frac P{2R}$ will give correct answer if ...
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3answers
30 views

Why's the base of the small arc length $r⋅dθ$?

As in this post, I'm trying to understand why area of a sector of a circle $= \dfrac{\theta r^2}{2} $ WITHOUT relying on $\dfrac{\theta }{2\pi} \pi {r}^{2}$ or $\dfrac{\theta }{2\pi} 2 \pi {r}$ or ...
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2answers
60 views

Ratio between the width of the intersection of two identical intersecting circles and radius, when the intersection is $\frac{\pi r^2}{2}$

Or more visually, if all sections of the below diagram were equal in area and the circles are identical, what is the ratio of s and r, or what is s in terms of r. I came up with an equation using ...
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2answers
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New Proof of Pythagorean Theorem (using inscribed circle)?

I was solving an easy problem for fun when I stumbled onto this, and was wondering if this was a correct and possibly a new proof of the Pythagorean Theorem. Given right triangle $\triangle ABC$, and ...
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0answers
19 views

Equation of A Circle, given: tangent line, line containing the center and radius [on hold]

How do I find the equation of the circle if it is tangent to the line 3x-4y+12=0 and the center is on x+4y=-1 its radius is 5
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4answers
32 views

Finding lengths within circles when the circles are tangent to each other [on hold]

I have been attempting this question about circle geometry, but not entirely sure how to go about it. I was initially thinking about using trigonometry, but couldn’t get anywhere with it. Any ideas ...
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2answers
98 views

Find $\left ( {C}' \right )$

Question: Circle $\left ( C \right ): x^{2}+ y^{2}- 2\,x+ 4\,y- 4= 0 \tag{1}$ in plane Oxy around point $O$ for $60^{\circ}$ and it maps in circle $\left ( {C}' \right )$ (find it!) First, we let $M(...
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1answer
54 views

Real life applications of a circle? (Conics)

for my Math 2U assignment, we have to discuss real life applications of different conic sections. However, apart from the wheel, I cannot find or think of any other real life applications of the ...
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2answers
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An identity associated with the centroid of a triangle

Four year ago, I am looking for a proof of my identity as follows: Let $ABC$ be a triangle, let $G$ be the centroid of $ABC$. Let $P$ be any point on the plane. Let $H, N, O$ on the plane such that: $...
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3answers
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What values gives the minimum area of the ellipse?

If the ellipse $\dfrac{x^2}{A}+\dfrac{y^2}{B}=1$ is to enclose the circle $x^2+y^2=2y$, what values of $A,B>0$ minimize the area of the ellipse? So far I've put the circle equation into the ...
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1answer
29 views

Calculating the break-even point between traveling around the inner / outer circle

Most days when I go to work, I ride a matatu (this is a Kenyan word. I am not in Kenya but the concept is the same). When I get off the matatu, I need to walk around the circle to get to the street ...
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2answers
61 views

Calculate the size of a polygon given the number and radius of circles that are on the perimeter

How can i calculate the size of a regular polygon if i have N circles of same size and all the circles must be on the perimeter of the polygon. Examples: 2 circles on hexagon 3 circles on ...
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1answer
799 views

Find coordinates for points on circle given R, 2 Points, and angle or 2 points and center?

I would like to find coordinates for points on a circle given: Radius of circle Coordinates of 2 points on the circle Angle of point 1, center, and point 2. Ultimately, I would like to write a ...
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1answer
13 views

Find value of x on a chord in the circle - which of the statements is true

in In the above image, is there a specific theorem that can be used to state which of the four statements must be true for x?
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1answer
56 views

Counting number of points making angle $< 90^\circ$

I have a around $1000$ points and $1000$ segments in the form of $(x_1, y_1, x_2, y_2)$ meaning the segment starts at coordinate $(x_1, y_1)$ and finishes at $(x_2, y_2)$. For each line i want to know ...
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2answers
110 views

What would be the area of this Red Marked points? And how to calculate this?

I have been given the length $L$ and the width $W$ of a rectangle and the radius $R$ of circle which is situated in the center of the rectangle . I need to find the area of the red marked portion. ...
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1answer
31 views

Radius of a circle containing circles.

If, inside a big circle, exactly $n (n ≥ 3)$ small circles, each of radius $r$, can be drawn in such a way that each small circle touches the big circle and also touches both its adjacent circles (as ...
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3answers
35 views

Cartesian product of $S^1$ and $S^2$

Can anyone help me imagine $S^1 \times S^2$? I understand that $T^2=S^1 \times S^1$, but I don't know what to do with the spheres. I am not even sure if it's in 3D. Thank you!
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3answers
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why do we use $\pi \cdot d/2+d$ for semicircles

I'm in grade seven and right now we are learning perimeter of a semicircle, and I noticed we use the formula $\pi\cdot D/2+D$ (where $D$ is the diameter) instead of $(\pi+2)D/2$, why do we do the ...
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1answer
21 views

reflections of a point around the sides of a triangle: the area stays constant

Given a triangle $\Delta ABC$ and a point $P$, we define $P_A, P_B, P_C$ as the reflections of $P$ around $BC, AC, AB$ respectively. Now, $P_A, P_B, P_C$ are collinear if and only if $P\in(ABC)$. (1) ...
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What is the center-point of a sphere called? [closed]

Based upon some quick searches I’ve discovered that the center of a sphere is literally called The Center of the Sphere. And the center of a circle is called the Circle’s Center. Have we really not ...
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1answer
32 views

Proving length of chords in a circle

$M$ is the midpoint of the chord $AB$ of circle $C(O,r)$. Show that if a different chord $ CD $ contains $M$, then $ AB < CD $ .(You may use Pythagoras's Theorem) Let $ \Delta CBM\sim \Delta ADM $...
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1answer
82 views

Maximum total area of n non-intersect circles?

Given n points on the x-axis, we give arbitrary radius for each point such that each constructed circle doesn't overlap another constructed circle from another point. Which means these circles do not ...
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2answers
34 views

Is it possible to cover all circle area with infinite lines starting from the center? [closed]

Is it possible to cover all area of a circle of radius r>0 with infinite lines starting from the center?
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3answers
239 views

Finding diameter of a circle using two chords and angle between them

Is it enough to find diameter of a circle using two arbitrary crossover chords with known length of each partition and angle between this two chords? If it's possible, how?
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3answers
49 views

Easier method of finding the equation of the circle circumscribing the triangle formed by 3 lines?

The equation of the circle circumscribing the triangle formed by the lines $y = 0, y = x$ and $2x + 3y = 10$ is? I know this can be done by solving two equations at a time and finding the vertex. ...
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1answer
27 views

A geometry problem about excircles and the tangency points

The excircle to side $BC$ of $\triangle ABC$ is tangent to lines $BC,AB$ and $AC$ at $D,E,F$ respectively. Let $P$ be the orthogonal projection of $D$ onto $EF$. Let the midpoint of $EF$ be $M$. Prove ...
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2answers
1k views

Plotting an arc with no center point - a practical solution please!

I need a mathematical solution to a very practical problem (laying a patio). The attached will hopefully explain. The center of the circle for the arc we wish to have is inaccessible (ie in the ...
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2answers
30 views

Locus of the point of contact of tangent.

Let $A, B, C$ be three points on a straight line, $B$ lying between $A$ and $C$. Consider all circles passing through $B$ and $C$. The points of contact of the tangents from $A$ to these circles ...
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3answers
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Finding the radius of the smallest circle that can circumscribe an equilateral triangle

Q:A puzzle board is in the form of an equilateral triangle that has an area of $7\sqrt{3}$ if the board is placed on a circular table, what should be the min area of the table so that the whole board ...
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0answers
28 views

Maximum number of nodes in a circle, with distance constraint

Given a circle $C$ with known radius $r$, I want to determine the maximum number of nodes in the circle, where there is a distance constraint between each two nodes equal to $s$, i.e. each two nodes ...
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1answer
821 views

From any arbitrary point $P$ on $y =\cos x$ tangents $PA$ and $PB$ are drawn to a circle which passes through

From any arbitrary point $P$ on $y =\cos x$ tangents $PA$ and $PB$ are drawn to a circle which passes through the points $(1,0)$ and $(3,0)$ and touches the circle $x^2+y^2-2x-8=0$ and have its ...
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2answers
118 views

Area of cut-out of three circles

Let three circles at different centers with different radii be given. They might intersect as shown in the picture. How to derive the area of the set (blue) in terms of the centers and radii? ...
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0answers
19 views

How to find the center of a circle moving and rotating in space having the coords of 3 points on the circle?

consider a circular plate with radius R , with three points on it at arbitrary coordinates on the plate area that form a triangle. Just imagine 3 holes on the plate area If I can have the new ...
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1answer
24 views

Chords in circle, finding angle [closed]

Circle image How can I find $\angle AOB$? (in terms of $\alpha$ and $\beta$)?
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0answers
12 views

Maximum relative quotient of radii of circles surrounding other circle touching neighbors exactly once? [duplicate]

Assume we have one circle with radius $R$ centered at origo $(0,0)$. How big circles of radius $r$ can we put around it touching it and each neighbor in one and only one point each if we want to fit $...
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3answers
37 views

find area of kite given side length

Circle with two tangent lines Above is the picture in question. A circle is given, center (-2,4) and a point outside the circle (0,10) is shown. Asked to calculate the area of the quadrilateral ABCD, ...