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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circle cuts three circles at the extremities of the diameter

If the circle $$x^2 + y^2 + 2gx + 2fy + c = 0$$ cuts the three circles $$x^2 + y^2 – 5 = 0\space;\space x^2 + y^2 – 8x – 6y + 10 = 0 \space;\space x^2 + y^2 – 4x + 2y – 2 = 0;$$ at the ...
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0answers
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$(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. Centroid of $\Delta ABC$ lies on $y=3x-4$, then the locus of $D$

$(x-1)(y-2)=5$ and $(x-1)^2+(y+2)^2=r^2$ intersect at four points $A,B,C,D$. If centroid of $\Delta ABC$ lies on $y=3x-4$, then what is the locus of $D$? I did try a couple of things, but I honestly ...
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4answers
47 views

How to calculate the radius of a circle inside a hexagon?

If I know how big is one side of a hexagon, what's the formula to calculate the radius of a circle inside it?
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2answers
29 views

Find a point on a circle given a point and height

Point on a circle Given : A point on a circle (point and angle), radius of the circle , height (Orthogonal to horizon ) I would like to find A point on a circle or , and The angle between the ...
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2answers
46 views

2 circles in an isosceles triangle

I've been given the following school problem: ABC is an isosceles triangle (AB = AC). The radius of the incircle is R and of the other circle (which is tangent to the incircle and to the legs of ...
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2answers
21 views

Proving the Secant Angles in the Circle

Ok, I know this is a very easy circle geometry problem, but I want to know that how to prove the theorem of angles in the circle. Like this image here: How can I prove that the angle $X$ is the ...
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1answer
23 views

calculate circle cardboard segments

I want to make a cardboard lamp, but i want it to look like half a sphere. Given a cardboard thickness of x, and a circle width of y, how many elements do I need and what radius do the elements need ...
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1answer
25 views

Question on circles…

If three circles with radii ${3}$,${4}$,${5}$ touch each other externally at points P,Q and R,then the CIRCUMRADIUS of ∆PQR is...?? My attempt i think that the let the point of the common ...
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1answer
19 views

find the equation of the diameter which passes through the origin.

I am given the equation of the circle $x^2+y^2−4x+6y=14$, and I am told to find the equation of the diameter which passes through the origin. However, I am unsure as to how to do this.
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1answer
26 views

What is the minimum radius $r$ of two intersecting circles that are spaced $x$ apart that completely enclose a square of length $w$?

Let's say we have two circles whose centers are spaced a fixed $x$ units apart from one another. Both circles have a radius $r$. Our goal is to identify the minimum value of $r$ so that the ...
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1answer
39 views

Proving three points lie on a same line

Given two circles $C1$ and $C2$ how do I prove that the line joining their centers will pass through the point of intersection of their internal common tangents.I tried to form a linear pair and prove ...
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1answer
35 views

Find length of a chord of a circle with radius $13$ cm given position of a point located on the chord.

A point located on a chord of a circle is 8 cm from one endpoint of the chord and 7 cm from the center of the circle. If a radius of this circle is 13 cm long, how long is the chord, in cm? Please ...
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3answers
73 views

Straight Edge - Only Geometric Construction

Given a circle, its diameter and a given point on the diameter, find a procedure to construct a line perpendicular to the diameter using only a straight edge. The perpendicular must pass through ...
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2answers
36 views

Intersection of two circles giving reversed answer

A little help needed. I need to derive the formula for the intersection points of two random circles. Equation: $\ x^2 + y^2 = 2.4^2$ and $ x^2 + (y+4)^2 = 17.16 $ I derived the equation which ...
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2answers
102 views

Construction using a straight edge only

Given a circle, its diameter and a point on the circle, find a procedure to construct a line perpendicular to the diameter using only a straight edge. The perpendicular must pass through the given ...
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3answers
36 views

How to check if two circles have common part?

I have equasion to calculate area of two circles with common part. Equasion common part But actually I just need to know if two cirlces have common part or no. Is there simpler equasion for that ...
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2answers
47 views

Power of a point proof

I found the question on page 13 of this link. Let $P$ be a point inside a circle such that there exist three chords through $P$ of equal length. Prove that $P$ is the center of the circle. I ...
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0answers
28 views

Proof related to power of a point

Let P be a point inside a circle such that there exist three chords through P of equal length. Prove that P is the center of the circle. I have read one proof related to power of point. It went like ...
2
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1answer
88 views

Construct a circle passing through a point $X$, which is externally tangent to two given circles

Given two disjoint circles $S_1$ and $S_2$, and a point $X$ external to both of them, is it possible to find the center of a circle that passes through $X$ and touches $S_1$ and $S_2$ tangentially, ...
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2answers
43 views

Geometry experts! Three equal tangential circles: What is the ratio of the blue line to the red line?

Consider the three tangential circles of equal radii inscribed in the equliateral triangle (linked to below). What is the ratio of the blue line to the red line? The red line is simply the diameter ...
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2answers
31 views

Prove $∠ADM = ∠ACB$ of triangle $ABC$ [closed]

Suppose that $ABC$ is a triangle. Let $D$ be its circumcenter and let $M$ be the midpoint of $\vec {AB}$. Show that $∠ADM = ∠ACB$.
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1answer
12 views

point on a circle extract dx and dy?

Given a circle like this, where i know the 2d coordinates of the center, and the radius of the circle, how do i determine the point on the circle, of more precise, dx and dy? illustration Sorry ...
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0answers
27 views

Prove a harmonic range from a familiar picture

During solving some simple problem (10th grade), I found this interesting problem, which I got no clue to solve it clean and properly. Hope someone can give me some hint to solve it. Thanks. Given ...
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2answers
32 views

Two circles touch internally. Find equation of smaller circle given equation of large circle

A circle C1 has the equation $(x+3)^2 + (y-2)^2 = 25$. Another circle C2 touches the first circle at a point P on the positive y-axis and passes through the centre of C1. The diameter of C1 is twice ...
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1answer
40 views

Find the other 2 interior angles of pentagon inscribed in a circle given 3 angles.

Given a pentagon $ABCDE$ inscribed in a circle with centre $O$. Three of the interior angles are $95^°$, $130^°$ and $138^°$. Find angle $x$ and $y$. I'm quite sure that $x$ and $y$ can be found as ...
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0answers
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About circles $n$ common chords

A circle of radius $\sqrt{5}$ and anotger circle of radius $\sqrt{10}$. Distance between two circles is $\sqrt{5}$. Please help me in finding the length of common chord.
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1answer
16 views

Compute mean on a torus / circular domain

I have $n \in \mathbb{N}$ values lying in a real, circular domain of period $T \in \mathbb{R}^{+*}$: $(\xi_i \in [0, T[^c)_{i \in \{1,\dots, n\}}$ I refer to the domain $C_T = [0, T[^c$ as a ...
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0answers
28 views

Drawing a straight line, and then curving along a circle

I am primarily a programmer, one with unfortunately relatively little education in mathematics, but I always try to get by. Right now, I am working with a simple set of rules. Draw a straight line in ...
2
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1answer
32 views

Prove that the center of a circle within a constructed triangles lies on the angle bisector

I was given steps to construct a figure: 1.) Construct a horizontal ray AB and a segment AC at an angle to the ray. Locate point D anywhere on ray AB and construct the segment CD. 2.) Construct the ...
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3answers
41 views

Is the closure of the set of all irrational rotation maps on $S^1$ dense in $Homeo(S^1)$?

I study about rotation maps on circle, and I have a question. Let $Homeo(S^1)$ be the set of all circle homeomorphisms with sup-metric $d(f,g)= \sup \{ d(f(x),g(x)| x \in S^1 \}$, and rotation map ...
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1answer
39 views

Proving that $P$ and $Q$ are symmetric in the line $XY$.

Let $ABCD$ be a cyclic quadrilateral with diagonals intersecting at $T$. Let $P$ and $Q$ be the projections of $T$ onto $AB$ and $CD$ respectively. Let $X$ and $Y$ be the mid-points of $AD$ and $BC$ ...
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0answers
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Harmonic inversion of an eccentric circle.

Inverted here is a circle with respect to another circle not as the conventional reciprocal inversion $ r_1 = \dfrac{a^2}{r_2}, $ but by means of a Lens formula known from time of Gauss: $$ 1/r_1 + ...
2
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1answer
41 views

Area of the shaded region of a circle

The parallelogram ABCD has a larger altitude of 4 cm and a shorter altitude of 3 cm. What is the area of the shaded region? The figure doesn't show to which side each of the altitudes are related, ...
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0answers
21 views

Prove Concurrency using Radical Axis of Circumcircles

Let the incircle of $\triangle ABC$ touch sides $BC,CA,AB$ at $D,E,F$, respectively. Let $\omega,\omega_1,\omega_2,\omega_3$ be the circumcircles of $\triangle ABCm,\triangle AEF,\triangle ...
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1answer
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Intersection of two circles in projective space

I have checked the existing question Intersection of two circles. and model for intersection of two circles in the complex projective plane - I do not think either of these answers my question. The ...
2
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1answer
48 views

Finding equation of a circle given three non - collinear points

A circle is given which passes through three non collinear points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ then prove that equation of this circle is given by $\begin{vmatrix} x^2+y^2&x&y&1\\ ...
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0answers
29 views

How many different ways can a circle intersect a triangle N ways?

Consider a circle intersecting a triangle. The circle and triangle can have between 0-6 total intersection points. Is there a mathematical formula for the number of possible ways they can intersect ...
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4answers
35 views

Project a point within a circle onto its edge. [closed]

What's the simplest way to find the intersection point of a straight line drawn from a circle's origin through a given point within the circle through the edge of the circle. I'm looking for the ...
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1answer
26 views

What are the coordinates for the center of the second circle? (Full question in body)

Full Question:A circle has its center at (6,7) and goes through the point (1,4). A second circle is tangent to the first circle at the point (1,4) and has one-fourth the area. What are the coordinates ...
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1answer
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Are each of the following statements for chords, radiuses, diameters and arcs of a circle true?

Are each of the following statements for a circle true? If a radius bisects a chord, then this radius is perpendicular to this chord. If a radius is perpendicular to a chord, then this radius ...
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0answers
42 views

Prove that the circle contains the polygon.

Given a convex polygon. The circle is constructed for every triple of consecutive vertices of the polygon.We get the n circles. Select the circle with the largest radius. Prove that the circle ...
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2answers
34 views

Prove that the centroid of $\triangle ABC$ lies on the circle $C_1$

Let $C_1,C_2$ be two concentric circles, the radius of $C_2$ being twice the radius of $C_1$. From a point $P$ on $C_2$ tangents to $C_1$, $PA$ and $PB$, are drawn. Prove that the centroid of ...
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2answers
83 views

Circle inversion of a circle

Given is a circle K with radius r and centre M1. K' is a second circle with radius r' and centre M2 that cuts K in two points A and B so that $[M1A]$ is orthogonal to $[M2A]$ and also $[M1B]$ is ...
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1answer
37 views

How can I find the radius R of a circle big enough to have n circles of different radii centered on its circumference, separated by an angle theta?

Sorry for the awful title, but this is a difficult problem to describe, so I made a picture. I want to find R given theta and all of the outer radii. Each of the outer circles must be centered on the ...
2
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1answer
19 views

All triangles that have the same orthocenter and circumcircle have the same nine-point circle

True or false? Prove it. I guess it would help to figure out whether 2 triangles can have the same circumcenter or orthocenter and not be congruent. I have no clue how to figure this out. If they ...
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0answers
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A new family circle associated with the Tucker hexagon and the Symmedian point

I am looking for the problem following: Let ABC be a triangle, let $A_1B_1C_1$ be a cevian triangle of the symmedian point. Let $B_aC_aC_bA_bA_cB_c$ be a Tucler hexagon of $ABC$. Such that $A_bA_c ...
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0answers
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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
228 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
73 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, ...