Questions tagged [inversive-geometry]
Questions related to Inversive Geometry and its applications.
82
questions
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1answer
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Which property of polar has been applied to this proof.
Which property of polar has been used please give its proof also.
1
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1answer
36 views
Inverse point of the center of a circle with respect to other circle
In a plane geometry class we were given a list of exercises that includes the following:
If a circle $L=(A,r)$ cuts the circle $K=(O,k)$ and $k^2+r^2=|OA|^2$, show that the inverse point of $A$ with ...
4
votes
1answer
75 views
Show that $A-$excircle is tangent to $(AST)$
Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of ...
1
vote
1answer
54 views
Is inversion logically consistent in neutral geometry? [closed]
A description of inversion can be found here.
Since the process uses points, circles, lines, extending lines, right angles, and similar triangles, it seems to me that this process could be proven to ...
5
votes
1answer
77 views
Doubt: Prove that the circumcircles of $\Delta ABC$ and $\Delta ADE$ are tangent with $\sqrt {BC}$
So, I recently started inversion and I have doubt in this solution . It's from "A beautiful Journey through Olympiad geometry " by Stefan Lozanovski. This Problem uses $\sqrt{BC}$
Here , I ...
1
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1answer
40 views
A question involving Self Polar Orthogonality
$\textbf{Question:}$ Let ω be a circle and suppose P and Q are points such that P lies on the pole of Q (and hence Q lies on the pole of P). Prove that the circle γ with diameter PQ is orthogonal to ...
3
votes
3answers
236 views
find a circle perpendicular to two other circles.
Here are two circles $C_1 = [ x^2 + y^2 = 1]$ and $C_2 = [ (x-2)^2 + y^2 = 3 ]$. The radii are $1$ and $3$ and the two circles are orthogonal to each other $C_1 \cdot C_2 = 0$. What is the equation ...
0
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0answers
32 views
Similarity coefficient of two inverted circles
The coefficient of similarity between two circles $C$ and $C’$ of radii $r$ and $r’$ is $$\frac{r}{r’}=\frac {k}{k_n},$$ where $k$ is the radius of inversion and $k_n$ is the square of the length of a ...
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1answer
58 views
Inversive geometry: is it possible to point-invert a Euclidean space and not produce a hyperbolic one?
Is it possible to point-invert a Euclidean space and not produce a hyperbolic one?
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0answers
43 views
How to determine if three distinct points $a,b,c \in \Bbb c$ are collinear using Mobius Transformation?
Given three points $\frac{3}{2} + i , 2i,-6+6i$. I have the mobius transformation that maps these three points to $0,1,\infty$ respectively as
$M(z) = \frac{(-4i+6)(z-(1+2i))}{(3-7i)(z-(10-20i)}$
...
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2answers
45 views
How to correctly determine the mobius transformation that maps three distinct points to 0,1,$\infty$
I was trying to find the mobius transformation that maps the point $i,-i,\infty$ to $0,1,\infty$. following the formula.
$M(z) = K \frac{z-i}{z+i}$. $M(i) = 1 \implies 1 = K\frac{i-i}{i+i} \implies \...
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1answer
95 views
Inversion in circle
Let $C$ be a circle with the middle point $O$ and the radius $r$, we say that the points $P$ and $P'$ are inverse points with respect to $C$ if:
$1.$ $|OP|·|OP'|=r^2$
$2.$ $P$ and $P'$ are on the ...
0
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3answers
843 views
inverse point with respect to circle
I'm reading Silverman's Complex variable with application.
at page 78, the author says
"We say that $s$ and $s^*$ are the inverse points with respect to circle in $\mathbb{C}$ if every line or circle ...
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0answers
26 views
Circular inversion notes or books
I am preparing a small lesson that I have to do about geometry. The topic of the lesson is circular inversion. Can you suggest me some sources where I can find some nice material about it?
Thanks!!
5
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7answers
386 views
What is the radius and center of the image of $|z|=1$ under $ f(z) = \frac{3z+2}{4z+3}$?
I would like to compute the image of the circle $|z|=1$ about the fractional linear transformation:
$$ f(z) = \frac{3z+2}{4z+3} $$
In particular, I'd like to compute the new center and radius.
...
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3answers
387 views
Circle whose radius is infinite
I have the intuition that a circle whose radius is infinite is a straight line.
Nonetheless, I don’t feel that what I’ve just stated is really scientific as it has some vagueness and lacks precision.
...
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0answers
91 views
Inversion in a sphere preserves circles (proof)
Inversion in the unit sphere, for a vector $x$, is defined by
$$\frac1x = \frac x{x^2} = \Big(\frac1{x\cdot x}\Big)x$$
How can we prove that a circle's inversion in the sphere is also a circle?
(I ...
1
vote
2answers
558 views
What can we do with inversive geometry?
I've recently been introduced to inversive geometry. This seems like it would be a very pretty area of study. Many sources that I have found seem a little old, however. I have two related questions:
...
2
votes
1answer
115 views
How to prove that the inversion $x \mapsto \frac x {\| x \|^2}$ in $\Bbb R^n$ sends circles to generalised circles?
Does anyone know how to prove (or have a reference for) the fact that the inversion
$$I : \Bbb R^n \setminus \{0\} \rightarrow \Bbb R^n \setminus \{0\}\\
x \mapsto \frac x {\|x\|^2}$$ sends circles ...
0
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0answers
36 views
What does “Inversion about $C$” mean?
Let $\Delta ABC$ be a triangle with orthocenter $H$ and
altitudes $AD, BE, CF$. Perform an inversion around $C$ with radius $\sqrt{CH \cdot CF}$. Where do
the six points each go?
I'm having trouble ...
5
votes
2answers
611 views
Proof of Miquel's six circle theorem
Theorem
Miquel's six circle theorem states that if in the following all cocircularities except the last one are satisfied, then the last one is implied.
In words: if $ABCD$ lie on a circle, and $...
3
votes
2answers
142 views
Circle Inversion
I don't understand the concept of circle inversion.
$OP \cdot OP' = k^2$
For example, in a circle $x^2+y^2=k^2$.
If I set a general point $P(x,y)$, why is its image $P'(\frac{xk^2}{x^2+y^2}, \frac{...
1
vote
1answer
53 views
Drawing a Poincelet Porism
How to draw a Poincelet porism?
In the above image, for the first case, we can draw the inner circle, then the triangle and then it's circumcircle.
But how to draw for the other cases and in general?
2
votes
2answers
54 views
Converse inversion
I was puzzling with circle inversions and was wondering about a converse of this:
Given 3 points A, B and M
Is there only one point R so that $d(R,A) * d(R,B) = d(R,M)^2$ or are there more of them?
...
0
votes
1answer
53 views
equation of inversion of a line [closed]
Say we have a line $L: ax+by+c=0 $ in the plane. Define the inversion of a point $(x,y)$ as $$\frac{(x,y)}{x^2+y^2}= (x',y')$$ with $$(x,y) \cdot (x',y')= 1.$$
What is the equation for the inversion ...
2
votes
0answers
93 views
(Logic behind) Geometric construction for imaginary roots of quadratic.
In the question on this site Geometric derivation of the quadratic equation
there are geometric constructions for real roots of a quadratic. As a follow-up
I found the construction below for ...
1
vote
0answers
126 views
Parallel lines as generalised circles in inversive geometry
Consider the inversion of circles ($w_1$ and $w_2$) w.r.t. the circle ($c$) that gives circles of infinite radius ($t_1$ and $t_2$ respectively).
What is the nature of the generalised circles $t_1$ ...
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2answers
429 views
Concentric circles in inversive-geometry
Is it possible to obtain two concentric generalised circles(parallel straight lines) through inversion of two circles w.r.t a third circle?
2
votes
1answer
107 views
Second point of intersection of two circles in Möbius geometry
Suppose I have two circles
\begin{align*}
(x-x_1)^2+(y-y_1)^2 &= r_1^2 \\
(x-x_2)^2+(y-y_2)^2 &= r_2^2
\end{align*}
and I also have one point
$$p=(x_p,y_p)$$
which is known to lie on both ...
0
votes
2answers
366 views
Find the image of vertical line $x=1$ under Mobius transformation of $\frac{2z}{z+1}$.
Let $f(z) = \dfrac{2z}{z+1}$ and $L_{3}$ = the line $x=1$.
I am required to find the image of $L_{3}$ under $f(z)$.
I have done this by choosing the three points $1, 1+i,\infty$ and seeing what they ...
0
votes
1answer
134 views
Finding the image of the x axis under a Mobius tranfsformation
Let $f(z) = \frac{2z}{z+1}$ and let $L_{1}$ be the $x$ axis.
I am told to show that $f(z)$ maps $L_{1}$ to itself.
I have picked three points to determine it image, $0,\ 1$ and $\infty$.
I'm ...
1
vote
1answer
93 views
Parallel lines in inversive geometry
Please give the definition of parallel lines(generalised circles) in inversive geometry?
Is there a unique definition (because i am confused about the duality of line with circle)?
If parallel is ...
0
votes
2answers
567 views
Finding the images under a Mobius transformation
Let $L1$ be the $x$-axis, let $L2$ be the $y$-axis and let $L3$ be the vertical line $x = 1$.
For each $k ∈ Z$ let $C_{k}$ denote the circle of radius $r = 1/2$ with centre $z = 1/2 + ki$.
Let $f(z) =...
5
votes
1answer
337 views
Is “square inversion” possible?
So, there exists in geometry circle inversion:
Can I perform a similar "inversion" technique through a square?
What would, for example, a square look like when inverted through another square?
2
votes
2answers
87 views
When does inversion respect vector addition?
A recent question inquired as to how one could characterize the solutions of the equation $\frac{1}{z_1}+\frac{1}{z_2}=\frac{1}{z_1+z_2}$ for complex $z_1,z_2$. This is trivially valid whenever ...
3
votes
1answer
320 views
Interesting circles hidden in Poncelet's porism configuration
This question is an investigation starting here, with a straightedge and compass construction of $ABC$ given $(R,r,h_A)$. The key lemma is the following one:
Let $\Gamma$ be a circle with centre $O$...
2
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0answers
128 views
Geometry - Inversion/Cross Ratios
Problem 5. Let ABCD be a given convex quadrilateral with sides BC
and AD equal in length and not parallel. Let E and F be interior points
of the sides BC and AD respectively such that BE = DF. The ...
2
votes
2answers
294 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 ...
0
votes
1answer
112 views
Centre of Invariant Circle under Inversion
Given an inversion of the plane, and a circle invariant under this inversion, what information do we know about the inverse of the centre this circle?
(I know that an invariant circle must be ...
2
votes
1answer
858 views
How is this circle inversion formula calculated?
I know about the inversion of a point inside a circle. But I was reading Peter Sarnak's paper on the Apollonian gasket, and got to the part where he was trying to prove descartes circle theorem. He ...
0
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1answer
319 views
Can someone explain this unit vector calculation for this circle inversion formula derivation?
I'm really stuck.
I'm learning about circle inversion. More specifically, I was trying to understand how to derive the inversion formula for a circle, which seems to be explained here.
http://...
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0answers
55 views
Let $I$ be an inversion and let $C$ be a circle such that $I(C)$ is also a circle. When do $C$ and $I(C)$ have equal radii?
Let $I$ be an inversion and let $C$ be a circle such that $I(C)$ is also a circle. When do $C$
and $I(C)$ have equal radii?
When it comes to inversion in a circle, I only know two cases:
a circle ...
1
vote
1answer
958 views
undefined angles with arcsin
I have this problem but I couldn't solve it. In a paper I'm reading for controlling a device, I need to generate the following angle
$$
\theta = \tan^{-1}\left( \frac{Y_{2} - Y_{1}}{ X_{2} - X_{1}} \...
4
votes
0answers
229 views
Prove special case of Brianchon's theorem using inversion
Brianchon's theorem says:
When a hexagon is circumscribed around a conic section, its principal
diagonals (those connecting opposite vertices) meet in a single point.
From interactive demo: (...
2
votes
1answer
128 views
Decide whether there exists an inversion that transforms one onto the other.
Question: For the following pairs of curves, Decide whether there exists an inversion that transforms one onto the other. Identify the inversion if it exists.
The circles $x^2+y^2=16$ and $(...
3
votes
1answer
486 views
Inversion: Find the transformation of a circle
To what point or curve does the inversion $I_{0,5}$ transform the
circle centered at (15,0) with radius 13?
($I_{0,5}$ means inverting on a circle of radius $5$ centered at $(0,0)$.)
I was looking ...
0
votes
1answer
143 views
If three circles have two common points, prove that every circle that is orthogonal to two circles is also orthogonal to third.
Three circles are given $k_1$,$k_2$,$k_3$ that have two common points A and B. Prove that every circle $k$ that is orthogonal to circles $k_1$,$k_2$, is also orthogonal to $k_3$.
Here is my proof ...
1
vote
2answers
94 views
inversion of the circle $t \mapsto (3 + is) + e^{it} $ around the unit circle.
We know that inversion interchanges lines and circles, but it's very hard to
The inversion map about the unit circle is just $\displaystyle z \mapsto \frac{1}{\overline{z}}$. As a Möbius ...
1
vote
3answers
134 views
On inversive geometry
I am given the following problem set:
Observe the circle $K$ with center $0$ and radius $r$ in the complex plane $ \mathbb{C} \simeq \mathbb{R}^2$. Show that the inversion on $K$ is given by the ...
0
votes
1answer
1k views
The radius of image of a circle under mobius transformation
A Mobius transformation of the plane takes $z \mapsto \frac{az+b}{cz+d}$. These are known to take circles to circles, but given an explicit circle, how do we compute the radius.
Let's parameterize ...
