All Questions
Tagged with circles analytic-geometry
450
questions
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40
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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 ...
- 602
0
votes
0
answers
18
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Writing the equation of a family of circles touching two circles
I know that the equation of a circle passing through the intersection of two circles is $S_1+\lambda S_2=0$ and I know that the equation of a circle passing through the intersection points of a line ...
- 767
2
votes
2
answers
103
views
Find the radius of the n+1th circle if it is constructed to be internally tangent to circle number 1, externally to circle number n and number 2.
Given a circle with radius $r_1$ a circle (number 2) with radius $r_2 < r_1$ is drawn such that it is tangent to circle internally. a circle (number 3) is constructed with radius $r_3= r_1 - r_2 $ ...
- 6,332
2
votes
1
answer
53
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How to calculate the radius of an outer circle from two symmetric internally tangent circles? (Applied Wind Turbine Problem 🔧)
Motivation & Context
The below problem is related to an open-source CAD model of a locally-manufactured small wind turbine.
The generator of the turbine is made from two rotating magnetic disks ...
- 23
1
vote
1
answer
47
views
Locus of centre of circle which bisects two smaller circles
Consider two fixed non-intersecting circles ( not necessarily of equal radii ). A circle which intersects both the circles also bisects their circumference. What is the locus of the centre of this ...
- 2,811
2
votes
0
answers
97
views
Finding the radius of the circle that is externally tangential to two given circles and the $x$-axis, without trigonometry [duplicate]
Given two circles:
$\color{red}{\Gamma_1: x^2+y^2=1}$
$\color{blue}{\Gamma_2: x^2+(y+\frac{1}{2})^2=\frac{1}{4}}$
$\color{green}{\Gamma_3: \dots ?}$
where $\color{green}{\Gamma_3}$ touchs $\color{red}{...
- 5,458
1
vote
2
answers
75
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Proving a trivial theorem involving a circle and a line
I want to prove that if the length of a line from the origin to the line $Ax + By + C$ is less than the radius of a circle with center $(0, 0)$, then the line $Ax + By + C = 0$ intersects the circle $...
- 172
0
votes
3
answers
188
views
Must three distinct spheres always intersect in exactly two points?
Two spheres A, B intersect in a circle, obviously, so a 3rd sphere C intersecting both A and B does so in 2 different circles.
It seems to me that the circle of the AC intersection must intersect the ...
- 331
1
vote
0
answers
85
views
Find equation of circles which are orthogonal to a system of coaxial circle.
I am doing an exercise on system of circles:
Find the equations of the circles which are orthogonal to all the circles of the system $x^2+y^2+2ax+2by-2\lambda(ax-by)=0\tag*{}$ where $\lambda$ is a ...
- 2,094
4
votes
3
answers
164
views
Deciphering the Ratio Challenge: Inscribe Circle in Quadrilateral $ABCD$
I trust this message reaches you well. I am reaching out to request your assistance in solving an intriguing geometry problem that I came across in a recent competitive exam. Despite my best efforts, ...
- 522
3
votes
2
answers
174
views
Simplify a formula with 449 terms - Radical circle
Context
The other day I wanted to answer this question.
Which is now closed so doesn't accept answers (but this isn't the important part).
Since I didn't know the topic I went to look it up but I ...
2
votes
1
answer
140
views
Locus of trirectangular tetrahedron [closed]
Let $A,B,C$ be points on the unit circle in the $xy$ plane, and $P$ be a point in space such that $PABC$ is a trirectangular tetrahedron with $P$ at the vertex. Find the locus of $P$ as $A,B,C$ vary ...
- 137
2
votes
1
answer
219
views
Geometry Challenge: Parallelogram in $\triangle$ $ABC$ with circumscribed Circle
I hope this message finds you in high spirits. I am writing to seek your expertise in solving a captivating geometry problem that I recently encountered in a competitive exam. Despite my best ...
- 522
5
votes
2
answers
110
views
Simple bisection geometry
Let $\triangle ABC$ have incenter $D$ and let the incircle intersect sides $BC,AB,AC$ at $E,F,G$ respectively. Extend $AB$ and $AC$ to meet the circumcircle of $\triangle ADE$ at $K$ and $I$ ...
- 109
0
votes
0
answers
13
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Help understanding possibly incomplete proof in coordinate geometry of one-parameter family of circles
If $S_1=0$ is a circle and $S_2 = 0$ is another circle and they intersect at two points- A and B, then the family of circles passing through A,B is represented as
$$S_1 + k*S_2=0$$
The proof in my ...
- 241