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.

4,053 questions
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Chords in circle, finding angle [closed]

Circle image How can I find $\angle AOB$? (in terms of $\alpha$ and $\beta$)?
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 \$...