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Questions tagged [circles]

For elementary questions concerning circles (or disks). A circle is the locus of points in a plane that are at a fixed distance from a fixed point. Use this tag alongside [geometry], [Euclidean geometry], or something similar. Do not use this tag for more advanced topics, such as complex analysis or topology.

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Place points on circle with constraints

Context Consider the following circle where each point on the circle is associated with a real in $[0,1)$: Problem I am looking for a mathematical function $f(x)$ defined on at least $\mathbb{N}↦\...
Flo's user avatar
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2 votes
0 answers
31 views

How to calculate the probability of a random great circle in a sphere pass through a certain spherical cap?

at some point, I got lost in some calculations. So, there is a sphere and a spherical cap in it. Imagine a great circle (one that passes through the center of a sphere) is chosen at random. I want to ...
Gabriel Farias's user avatar
2 votes
1 answer
71 views

concyclicity related to the Humpty point

$AD$ and $CE$ are the altitudes of triangle $ABC$, $M$ is the midpoint of $AC$. Circumcircles $\omega_1$ and $\omega_2$ of triangles $AEM$ and $CDM$, respectively, intersect at point $P$. $CP$ and $AP$...
Meison's user avatar
  • 307
0 votes
0 answers
25 views

Rational quantities associated with a bicentric heptagon

For odd reasons of my own, I would like to create a math puzzle involving a bicentric heptagon (i.e., a 7-sided polygon which has both an inscribed and a circumscribed circle). The puzzle is to be ...
tuna's user avatar
  • 547
-1 votes
0 answers
28 views

Apeirogon equal circle? [closed]

Apeirogon is the same circle? My question is, can you make sure that a line is created from a set of dots? If this is possible, isn't it true that the circle is polygonal? If so, is the curve made up ...
이사무엘's user avatar
-1 votes
0 answers
20 views

Find the condition that the Lx + my + u = 0 should be a normal to the circle x^2+y^2 +2gx + 2fy + c = 0 [closed]

What is the condition so that line is normal to circle. Which formula should be used in the question.
Sujal Panjiyar's user avatar
2 votes
2 answers
79 views

How to calculate intersection points of diameters with an inner circle?

I am working on a visualisation project where I am attempting to create a circle illusion using moving dots along the diameters of a larger circle. To achieve this effect, I need to calculate the ...
Ibrahim Khalil's user avatar
0 votes
0 answers
19 views

Number of chords to partition concentric circles to exclude area

Sorry for the poor title, but geometry is not my strong suit. Nerd sniped myself with this problem while masking off circular vinyl negatives on the bottom of circular glasses for etching. Is there a ...
mascoj's user avatar
  • 145
-4 votes
0 answers
24 views

PC extended intersects the circle at D. Q is a point on CD such that ∠DAQ = ∠PBC. Show that ∠DBQ = ∠PAC [closed]

Refer to the diagram below. A, B, C are points on the circle. PC extended intersects the circle at D. Q is a point on CD such that ∠DAQ = ∠PBC. Show that ∠DBQ = ∠PAC
Anshika Kansal's user avatar
0 votes
0 answers
48 views

Elliptical Grid Mapping in Shader

I wanted to make a Elliptical Grid Mapping Shader, but it is not a perfect square and it is rotated. If i multiply the coords by sqrt(2.) and divides them after again, it is an square, but still ...
Taxy's user avatar
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0 answers
63 views

Prove centre is inside cyclic quadrilateral with perpendicular diagonals

Let $ABCD$ be a convex cyclic quadrilateral, and the diagonals $AC$ and $BD$ are perpendicular. The circumcircle of $ABCD$ has centre $O$. I am trying to prove that the centre $O$ is inside $ABCD$. I ...
wenbang's user avatar
-2 votes
0 answers
26 views

Number of parameters needed to find a point on $S^n$

Firstly, let me point out that the following argument can be easily extended to $S^n$ for every natural number $n$, so I will just focus on $S^1$. Consider the circumference $x^2+y^2=1$, centred at $O=...
Davide Masi's user avatar
2 votes
1 answer
56 views

Finding the circumradius of a cyclic hexagon, given three non-consecutive sides and the fact that the midpoints of all sides are also cyclic

Cruel Geometry Question, Mock AIME-i 2015: Let $A_1A_2A_3A_4A_5A_6$ be a hexagon inscribed inside a circle of radius $r$. Furthermore, for each positive integer $1 \leq i \leq 6$ let $M_i$ be the ...
CLASH ROYAL's user avatar
3 votes
2 answers
62 views

How to interpret the condition of a circumference rolling without slipping on another circumference

Suppose a circumference of radius r and center $\Omega$ rotates with constant angular velocity $\omega_D=\dot\phi e_3$ (D stands for disk) around an axis parallel to $e_3$ through $\Omega$. Let $\...
Davide Masi's user avatar
8 votes
1 answer
135 views

Double Contact Chained Ellipses Problem

A few years ago, when I played around with GeoGebra, I came up with the following conjecture. Conjecture Let $n\in\mathbb{N}, n\geq3$. Let $E$ be an ellipse, and let $E_{1}, E_{2}, \dots, E_{n}$ be ...
K. Miyamoto's user avatar
2 votes
1 answer
101 views

The area of a inscribed polygon tends to the area of the circle

As it is broadly known, given a circle of radius $r$, its area is equal to $\pi \cdot r^2$. My goal is to prove this formula using inscribed polygons. Let´s call $n$ the number of sides of a regular ...
IkerUCM's user avatar
  • 402
0 votes
1 answer
80 views

Problem involving 3 circumferences [closed]

I am trying to prove the statement $OQ+QP=2OP$ where the letters, the draw and my attempt is in the following image. Now, as you can see I am not on the right track because after substituting I do not ...
MiguelCG's user avatar
  • 345
1 vote
0 answers
42 views

How do you find the intersection between 2 complex shapes?

One thing I've done to find if two perfect circles intersect is to compare the radii of both circles to the distance between them, which isn't complicated at all. However, for non-circle shapes, I don'...
ImSomeOtherGuy's user avatar
8 votes
0 answers
162 views

For what values of $n$ can coins of radius $\frac12,\frac13,\frac14,...,\frac1n$ be held rigidly in a circular tray of radius $1$?

For what values of $n$ can circular coins of radius $\frac12,\frac13,\frac14,...,\frac1n$ (at least one of each, and no other kind of coin) be held rigidly in a circular tray of radius $1$? By "...
Dan's user avatar
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69 votes
3 answers
7k views

Geometry question about a six-pack of beer

On a hot summer day like today, I like to put a six-pack of beer in my cooler and enjoy some cold ones outdoors. My cooler is in the shape of a cylinder. When I place the six-pack in the cooler ...
Dan's user avatar
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14 votes
3 answers
850 views

Sangaku problem involving eight circles

I made the following sangaku problem. $\dfrac{\text{Area of the orange circle}}{\text{Area of a blue circle}}=\space ?$ Description of diagram. In this question, circles of the same color are ...
Dan's user avatar
  • 25.8k
2 votes
0 answers
33 views

What is the maximum area of n non-overlapping equal area triangles inscribed in a circle of radius

What is the maximum area of n non-overlapping equal area triangles inscribed in a circle of radius 1? For n = 1, the triangle is equilateral. For n = 2, we have 2 isosceles right triangles sharing a ...
Ultima Gaina's user avatar
0 votes
0 answers
32 views

If I have x and y coordinates of a point along the arc, how do I convert that to a percentage of PI?

I am using Javascript to create shapes in canvas. I am creating an arc where you specify the start and end angle of the arc to show where along a circle the arc begins and ends. They are initially set ...
Jared H's user avatar
0 votes
1 answer
38 views

Proving Symmedian intersects intersection of tangents

I'm going through Evan Chen's "Euclidean Geometry in Math Olympiads" and I've come to Chapter 4's section on Symmedians. Proposition 4.24 says: Let $X$ be the intersection of the tangents to ...
PabloGamerX's user avatar
0 votes
0 answers
76 views

Lattice points inside $x^2+y^2=r^2$

In the interior of the square of side length $N$, whose two corners are lattice points $(m,m)\in\Bbb Z^2$, there are $(N-1)^2$ lattice points. So, in the intetior of the circle $$x^2+y^2=r^2,\,\, r\in\...
Bob Dobbs's user avatar
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0 votes
0 answers
23 views

The unit disc contains finitely many dyadic squares whose total area is arbitrarily close to the area of the disc

Exercise 1.25.a in Pugh’s Real Mathematical Analysis states that Given $\epsilon > 0$, show that the unit disc contains finitely many dyadic squares whose total area exceeds $\pi - \epsilon$, and ...
user avatar
7 votes
1 answer
202 views

A beautiful property that connects five points located on a circle to the nine point circles

A while ago, while playing with the GeoGebra program, I came across a distinct geometric property, and I would like to know whether it has been previously discovered or not, and also how this complex ...
زكريا حسناوي's user avatar
13 votes
6 answers
556 views

Is there a geometrical diagram in which it is evident that two circles' radii have ratio $1:11$?

There are geometrical diagrams in which it is evident* (to a skilled geometer) that two circles' radii have a certain integer ratio. For example, for the following diagram, it is evident that the ...
Dan's user avatar
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0 votes
1 answer
43 views

Prove that the tangents from point $A$ and the tangent from point $B$ to the circle $k_1$ are parallel.

Given is a circle $k$ with diameter $AB$. On it, we choose a point $M$, which is not coincident with $A$ or $B$. Let $k_1$ be the circle that has its center at $M$ and is tangent to the diameter $AB$. ...
user avatar
0 votes
1 answer
77 views

Find the segment BT in the triangle inscribed below

In the figure, $AB.BC = 60$ and $BT.TP = 40$. Calculate BT with B and T tangency points. (Answer:$2\sqrt5$) I try: $AT.TC = BT.TP \implies AT.TC = 40$ $AM.AB = AT.AC$ $AT^2 = AM.AB \implies AT^2 = AT....
peta arantes's user avatar
  • 7,031
12 votes
3 answers
365 views

Sangaku problem about three chords in a circle

Inside circle $O$ (shown with white interior), a blue circle sits at the bottom. $O$ has two chords that meet at the top of $O$ and are tangent to the blue circle. $O$ also has a horizontal chord ...
Dan's user avatar
  • 25.8k
1 vote
1 answer
103 views

Parametric Equation of a unit circle when the angle between $x$-axis and $y$-axis is not $90$ degrees

I know in regular Cartesian coordinates the parametric equation for a unit circle is $x=\cos(\theta)$, $y=\sin(\theta)$, and if the $x$ coordinates are stretched by an amount $a$, and the $y$ ...
Anders Gustafson's user avatar
14 votes
1 answer
336 views

Sangaku problem: Show that five circles have equal radii.

Inspired by Sangaku problems such as these, I created the following Sangaku problem. Consider the following diagram. Description: The diagram shows the circle $x^2+y^2=1$, two red chords with ...
Dan's user avatar
  • 25.8k
0 votes
0 answers
20 views

Question on circle and binomial coefficient [duplicate]

On a circle C, put n points on the circumference of C. Now the question says that 𝐶ₙ is the number of chords (such that no 3 chords intersect at any one point), 𝑖ₙ that of intersections of chords ...
Moksh Singh Dangi's user avatar
4 votes
4 answers
269 views

I need help in understanding the alternative solution provided to solve this geometry question of calculating area of quadrilateral

Question: Solution provided: I understand this part that equal chords of a circle subtend equal angles at the center, but after this the faculty transformed this whole diagram to one shown below in ...
Vasu Gupta's user avatar
  • 1,050
10 votes
2 answers
506 views

A circle is inscribed in a triangle, with three other circles in the corner regions. The radii are integers. Possible values of the largest radius?

Consider four circles with integer radii inscribed in a triangle as shown. That is, a circle with integer radius $R$ is inscribed in a triangle, and three other circles with integer radii $a,b,c$ are ...
Dan's user avatar
  • 25.8k
4 votes
3 answers
187 views

Find the sum $PC^2+PD^2$ in the trapezoid inscribed below

If there is a semicircle of diameter $AB$ in which an isosceles trapezoid $ABCD$, ($AB \parallel CD$) is inscribed. On $AB$, we take a point "$P$" such that $PA^2 + PB^2 = 5^2$. Calculate: $...
peta arantes's user avatar
  • 7,031
1 vote
1 answer
46 views

Finding maximum Area of a convex quadrilateral inside a unit circle

Here's a question of my own: • You have a unit circle. • You want to place two points inside the circle. • Now from each point you connect them to the nearest point on the circumference. (Now you have ...
Prim3numbah's user avatar
2 votes
2 answers
113 views

Can you give me an intuative understanding of the measure of curvature: "4 1/mm"?

Update: Follow up questions: Why is the reciprocal of the radius deemed more descriptive of the curvature that the radius itself? Who uses these 1/mm (or I assume 1/m or even 1/") measures of ...
Buk's user avatar
  • 147
3 votes
1 answer
53 views

In a triangle ABC : 2 externaly tangent circles, also tangent to BC with centers on line segments AB and AC : envelope of their lines of centers?

The figure here gives an illustration of the configuration described in the title in 4 cases ; consider especialy the fourth one, materialized by red circles, red center points, and a red line segment ...
Jean Marie's user avatar
  • 83.9k
1 vote
0 answers
40 views

How to prove that 3 circles are concurrent? [closed]

I have come across this question in Sharygin 2024 - Points A′ , B′ , C ′ are the reflections of vertices A, B, C about the opposite sidelines of triangle ABC. Prove that the circles AB′C ′ , A′BC′ , ...
MathsGuy's user avatar
4 votes
2 answers
70 views

Ratio of radii in rings of tangential circles [closed]

The picture of my table mat above is the inspiration for this problem. Suppose we have a central circle. Around this central circle we lay a ring of $n$ smaller circles of radius $r$ such that they ...
Cristof012's user avatar
4 votes
3 answers
147 views

Can the center of circumscribed circle in a triangle be on the incircle?

Besides the obvious answer of an isosceles right triangle, can there be other triangles where the center of its circumscribed circle is located on the perimeter of its incircle? I tried using the ...
Sadra Daneshvar's user avatar
1 vote
1 answer
35 views

What is the relationship between the silver ratio and the postion of a circle in the corner of a triangle?

I was recently trying to figure out how much to offset a circle in the corner of a right-angled triangle and found empirically that the x-offset needed to be around 2.414 which I later found to be the ...
TyghtMo's user avatar
  • 13
0 votes
2 answers
114 views

Four Circles radii

Consider following constellation of four adjacent circles. Question:(Initial question doesn't give an unique solution; see edit) Assume we know the radii $R_1,R_2,R_3$. Is there a geometric/synthetic ...
user267839's user avatar
  • 7,589
0 votes
1 answer
36 views

In concentric circles Triangle formed from intersection of a line making 45 degree with x axis where inner circle meets x axis to outer circle [closed]

I have two concentric circles one of radius 5 cm and outer one of 10 cm, their centers being 0,0 I want to calculate P B and H of the triangle formed by intersection of a line on outer circle making ...
Ken Kaneki's user avatar
4 votes
1 answer
227 views

A geometry problem regarding the relationship between the circumradius and the radii of three other triangles [closed]

I came across this problem in my geometry book and still cannot find the solution. I am strugling with it for over a week now and I kind of ran out of ideas how to approach it. So some hints would be ...
Dmitro's user avatar
  • 67
0 votes
0 answers
40 views

Chords $\overline{AB}$ and $\overline{CD}$ of a circle meet at the point $E$ outside the circle. Prove that

(a) $\angle A\cong\angle C$ (b) $\angle1\cong\angle 2$ (c) $\triangle ADE$ and $\triangle CBE$ are equiangular. I tried upto some extent but did not know I to solve this problem. I know that length ...
Noor Aslam's user avatar
0 votes
2 answers
173 views

Calculating circle offset for each angle when reference point is not circle center

I try to calculate offset change at outer rim of a circle for each angle with respect to C1 but can not figure out. its is easy on the paper with geometry but very difficult to formulate it. The ...
ebbac44's user avatar
  • 11
3 votes
1 answer
132 views

A geometry problem involving a tangent of circumcircle at a vertex of triangle

This is a problem in a geometry test, which I'm using to practice, however I can't solve it so I decided to post it here, to (at least) find a hint, or a solution (if possible). Let $ABC$ be an acute ...
anonimo's user avatar
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