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Questions tagged [circle]

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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1answer
12 views

Proove Line Circumscribed Circle; Incircle; Excircle

How can I proove, that the circumscribed circle of a triangle does exactly cross the middle of the line that goes from the incenter of the incircle of the triangle to the excenter of the excircle of ...
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1answer
36 views

Approximation of the quadratic formula with straightedge and compass

Given a directrix and a focus (blue), we can define a parabola as illustrated below. We suppose the parabola intersecting the $x$-axis in correspondence of the red dots. We draw the line ...
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2answers
68 views

Finding the side length of an equilateral triangle having three inscribed $120^\circ$ sectors in a certain arrangement

How do i even start this question? I thought of using length of tangent equal from exterior point but still of no use.
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0answers
29 views

Is there any Geometrical name for the overlapping region of three circles (at least two should overlap)?

Is there any geometrical name for the region depicted in the attached picture?
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1answer
74 views

Show that $BP+BQ=2PQ $

Let consider a circle of diameter $CA $ and $B\in CA $ such that $A\in [CB] $ and $AB=\frac {CA}{2} $. If $M \in [CA] $ such that $AM=\frac {CA}{3} $ and $P, Q $ on circle such that $P, M, Q $ ...
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2answers
64 views

Prove that $|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2$. [on hold]

A polygon $A_{1}A_{2}...A_{n}$ has a circumscribed circle with radius $R$. Prove $$|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2.$$
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1answer
47 views

Prove that $NK$ is tangent to the circumcircle of $\Delta KEF$ .

Consider a circle $O$ with radius $R$. $ABCD$ is cyclic quadrilateral, the intersection of $AC$ and $BD$ is $K$. $P$ and $Q$ are respectively the midpoints of $KD$ and $KC$. The intersection of $AP$ ...
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1answer
14 views

Under what conditions do the angle bisectors of a triangle make $120^\circ$ angles at the incenter?

I haven't messed around with geometry in a while. What criteria does this triangle need to meet so you can state that the incircle angles $\alpha,\beta, \gamma $ here equal 120º? Do any of the ...
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1answer
35 views

Find the area of an ellipse inside a circle [on hold]

I have a cellular signal calculation function, which calculates the signal given the distance from the antenna. Without the constants, the function is basically: $f(d)=1/(d^α)$ where α is a parameter. ...
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3answers
608 views

Rational points on a circle with centre as $(\pi,e)$

What is the number of rational points on a circle having centre as $(\pi,e)$?
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0answers
32 views

How was trigonometry used to define this racing line [closed]

I was searching through the internet for an equation that defines the path followed by a racing line, when I came across the following formula on StackExchange posted by another user as a reply for ...
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0answers
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Concept of angle with circle

Has the concept of angle originated from circles ? I don't understand the concept of angles. I want to know how people start using the word "angle" in mathematics. When I want to know about angles, ...
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2answers
65 views

Show that $OX=OY$.

Let $O$ the circumcenter of an acute triangle $ABC$. Let $\alpha $ the circle trough $A $ and $B $ tangent to $[AC] $, and $\beta $ the circle trough $A, C $ tangent to $[AB] $. A line trough $A $ ...
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24 views

How many regions are made when to set of circles intersect?

We are given four parameters to draw some circles which are centered on the x axis: center of the circles(x1 and x2),both are located on the x axis the maximum radius(k1 and k2) this is how we use ...
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1answer
28 views

Completing the square to find centre and radius of a circle

$x^2 + y^2 -2x - 5y + 16 = 0$ So I went like this: $$x^2 - 2x + 1 - 1 + y^2 - 5y + \frac{25}{4} - \frac{25}{4} = - 16 \\ (x-1)^2 + (y-\frac{5}{2})^2 = -\frac{64}{4} + \frac{25}{4} + \frac{4}{4}$$ I ...
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3answers
40 views

Find angle of line along a circle's perimeter? [duplicate]

If I know coordinates of point $A$, coordinates of circle center $B$ and $r$ is the radius of the circle, is it possible to calculate the angle of the lines that are passing through point A that are ...
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1answer
28 views

Proof that the circumcenters of sub triangles forms a triangle congruent with the original triangle

Let $\triangle ABC$ be a triangle with orthocenter H and let $O_A, O_B, O_C$ be the circumcenters of triangles $\triangle BCH, \triangle CAH, \triangle ABH$, respectively. Prove that the $\triangle ...
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1answer
33 views

Number of roots of unity on the unit circle

I have to find the number of roots of unity on the unit circle |z|=1 in the argand plane. I know that there are n roots of the the equation $z^n=1$ and all of them lie on the given circle. Does that ...
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1answer
55 views

Find a line along a circle's perimeter?

If I know coordinates of point $A$, coordinates of circle center $B$ and r as the radius of the circle, is it possible to calculate lines that are passing through point $A$ that are also tangent to ...
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1answer
32 views

Find the radius of circle $C$ that touches the parabola $y= \frac 12 x^2$, at point $(2,2)$.

The circle is contained in the domain $x \geq 0$ and $y \geq 0$ and the parabola equation is $y = \dfrac{1}{2}x^2$.
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2answers
71 views

Are $M, P, N $ collinear?

Let $\alpha $ a circle of diameters $AB $ and $\beta $ a circle tangent to $AB $ in $ C$ and tangent to $\alpha$ in $T $. Let $M\in \alpha $ and $N\in CB $ s.t. $MN\perp AB $ and $MN $ is tangent ...
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0answers
51 views

Show that $\angle AMC=\angle CMN $.

Let $\alpha $ a circle of diameters $AB $ and $\beta $ a circle tangent to $AB $ in $ C$ and tangent to $\alpha$ in $T $. Let $M\in \alpha $ and $N\in CB $ s.t. $MN\perp AB $ and $MN $ is tangent ...
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0answers
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Shortest closed Steiner chain of circles sandwiched between circumcircle and incircle of triangle

Consider Steiner chain of circles with the external circle of radius $R$, the internal circle of radius $r$ and $n$ circles in a chain with the radii $r_1,\dots,r_n$. Known condition for the ...
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1answer
26 views

Show that the center of the circle $(ABC) $ is on $AE $.

Let $\triangle ABC $ and $D\in [BC] $ s.t. $\angle BAD=\angle DAC $. Let $BE\perp AD $ where $E $ is on the circle (ABD). Show that the center of the circle $(ABC) $ is on $AE $. I have no idea ...
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2answers
25 views

How to draw the equation $x = r\cos(\theta)$ and $y = r\sin(\theta)$

Studying for calculus right now and in the notes it says a circle can have the equation x^2 + y^2 = r^2. ...
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0answers
16 views

Functions of perimeter and area of parts from two circles

We have a circle with radius $1$. Next we draw a circle with radius $r$, whose center lay on the first circle, so $0\leqslant r\leqslant2$. When $r=0$ there is only $1$ part with $$C(0)=2\pi, S(0)=\pi$...
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2answers
171 views

A conjecture related to any triangle

Given one side $AC$ of any triangle $\triangle ABC$, we can draw the couple of circles with center in $A$ and passing through $C$ and with center in $C$ and passing through $A$, obtaining two points $...
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5answers
37 views

Max and min of complex locus $|z-2-i|=1$

I do not understand the process that is required in order to find the max and min values of $|z|$ in $|z-2-i|=1$. My textbook implies to use inspection which I find a bit confusing. I tried to ...
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2answers
71 views

can someone help in solving this triangle-circumcircle question? [closed]

$AB$ is a chord of a circle and the tangents at $A$, $B$ meet at $C$. If $P$ is any point on the circle and $PL$, $PM$, $PN$ are the perpendiculars from $P$ to $AB$, $BC$, $CA$. Prove that $PL^2= PM\...
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1answer
62 views

A novel (?) construction of the regular pentagon with straightedge and compass

With reference to the triangle $\triangle ABC$ illustrated in the picture below, given the side $AC$, the five points $B,D,E,F,G$, in the conditions discussed here, determine a circle (red). Let us ...
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2answers
19 views

Number of regions [closed]

what is the max no of closed regions that can be formed with drawing four lines inside a circle? Correct answer is $11$ but I'm getting $8$.
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0answers
22 views

How to find the largest regular shape inside irregular convex polygons in 2D

I want to find a way that can find a regular or common shape (e.g a circle, triangle, square or rectangle) that has the maximum area inside different irregular 2D polygons. From this article, it ...
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2answers
29 views

Circle through point of intersection of two circles.

Suppose we have two circles $S_1$ and $S_2$. Why is the circle through the point of intersection of the two circles given by: $S_1 + \lambda(S_2 - S_1) = 0$ and not: $S_2 + \lambda(S_1) = 0$? ...
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1answer
19 views

Proving a circumference is not homeomorphic to two adjacent ones

Let $X$ be the unit circle in $\mathbb{R}^2$; that is $X=\{(x,y):x^2+y^2=1\}$ and has the subspace topology. Consider $Y$ to be the subspace of $\mathbb{R}^2$ given by $Y=\{(x,y):x^2+y^2=1\}\cup\{(x,y)...
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2answers
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How to prove that AE is the arithmetic mean of AB and AC

The internal Bisectors of angle A of triangle ABC meets the circumcircle at D. If DE,DF are the perpendiculars to AB , AC respectively from D,then prove that AE=(AB+AC)/2
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1answer
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Pizza Problem, percents

A medium size pizza at Ristorante Porcupine is 10 inches in diameter. A large pizza is 21 inches in diameter. What percent larger is the area of a large pizza? Express your answer to the nearest ...
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1answer
61 views

Locus of a vertex defining a circle intrinsic to a triangle

Given a triangle $\triangle ABC$, we draw the circles with centers in two vertices (say, $A,C$ in the picture below) and passing through the third one (say, $B$), determining the points $D$ and $E$ on ...
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2answers
33 views

Tangential Circles

For a natural number $n$, the circles $C_{3n-2}$, $C_{3n-1}$, and $C_{3n}$ have the same radius, tangential to each other externally, and tangential to the circle $C_{3n+1}$ intenally. If the radius ...
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1answer
25 views

Radius of the smaller circle

Two circles $(C_1$ with radius $r_1$, and $C_2$ with radius $r_2$) are having the same centre. $P_1$ and $P_2$ are two distinct points lying on the circumference of $C_1$. The length of line $P_1P_2$ ...
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0answers
19 views

How to find the circumscribed circle for given set of circle? [duplicate]

I have $n$ circles, their centers $(x_i,y_i)$ and radii $r_i$ are known, $i=1,2, ...n$. I need to find the circle (red line in the figure below) in which the mentioned circles above are placed. $...
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4answers
76 views

Largest circle between lines through point

Calculate circle through a point, with its center on a line I am trying to make a program in java that calculates the largest circle between two tangential lines through a point that is not on one of ...
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1answer
32 views

Calculation the location of Ghost Ball

Regarding to scheme as follows, when i move Guiding Line (Purple Colored), Ball A (Ghost) will slide around Ball B on red Axis. For this reason, i need to find β angle to calculate the exact location ...
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1answer
37 views

Length of a shadow cast by an object on a sphere

Say there is an object with height y standing on a spherical globe with radius r. A light ray casts a shadow from the object to the ground at angle θs. How can I find the length of the shadow d that ...
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0answers
151 views

Can The Existence Of $\pi$ Be Proved Without Formal Analysis?

I hope this question is not too long, but I have included some extra information to clarify the context of the question and hopefully avoid the 'circular' arguments which inevitably occur on this ...
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4answers
81 views

Complex Number solutions for a Circle

I have a circle of radius 5 with its center at the origin represented as $X^2+Y^2=25$. I get that it has a solution for all values ranging from $-5$ to $+5$. My question is what does it mean when the ...
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2answers
57 views

A Question on circles based on co-ordinate geometry.

The line $4 x + 3 y – 4 = 0 $ divides the circumference of the circle centred at $ (5,3)$ in the ratio $1:2$ . Then the equation of the circle is? I tried to solve this problem my taking parametric ...
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23 views

Circular geometry problems

enter image description here How do i find x - y? Can someone help me to solve? Circular geometry problems seems hard to me. Thanks!
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1answer
24 views

The distance between the centers of two circles K1 and K2 is 2cm. Determine the surface area of the figure that belongs to both circles.

The distance between the centers of two circles K1 and K2 is 2cm. They intersect under a right angle, the radius of K1 is 1cm. Determine the surface area of the figure that belongs to both circles. [ ...
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1answer
24 views

Draw a line from point to circle circumference, but pointing at the center

I have an app, where I draw a graph. From each circle in this graph there are some lines to other circles. To not mess up my drawing, I tend to draw the line up from the circle, then horizontally in ...
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1answer
35 views

Prove that $P,I$ and $C$ are collinear

The incircle of $ABC$ touches $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. The line through $A$ parallel to $LK$ meets $MK$ at $P$. Show that $ \angle API = 90$ and that points $P,I$ and $C$ are ...