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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3answers
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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
48 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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1answer
89 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
47 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
32 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
14 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
29 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
34 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
44 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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1answer
17 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 ...
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1answer
34 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 $...
3
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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
14 views

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 + 1/...
2
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1answer
51 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
22 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 BDF,\...
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1answer
19 views

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
votes
1answer
52 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\\ x_1^...
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0answers
36 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
37 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
27 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
55 views

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
45 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 ...
2
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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 $\...
2
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2answers
89 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 ...
2
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1answer
39 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
21 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
14 views

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
244 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
118 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
66 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, $...
2
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1answer
36 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
87 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
78 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
80 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 I ...
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1answer
156 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 ...
0
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1answer
61 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 $V$...
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2answers
54 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 $|AB|=|AC|...
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0answers
56 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 ...
2
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2answers
99 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
71 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 ...
0
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1answer
48 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 \pi}{...
3
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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
37 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
45 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
70 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 ...