Questions tagged [inversive-geometry]

Questions related to Inversive Geometry and its applications.

Wikipedia says

In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion. These transformations preserve angles and map generalized circles into generalized circles, where a generalized circle means either a circle or a line (loosely speaking, a circle with infinite radius). Many difficult problems in geometry become much more tractable when an inversion is applied.

The concept of inversion can be generalized to higher dimensional spaces.

$\newcommand{\inv}{\operatorname*{inv}}$ Inversion through the sphere $S(c,r)$ centered at $c$ with radius $r$, is defined as $$ \inv_{S(c,r)}(x)=c+r^2\frac{x-c}{|x-c|^2} $$ That is, the direction from $c$ to $x$ is preserved, but the new distance from $c$ to $x$ is $r^2$ divided by the old distance. Thus, $S(c,r)$ is preserved by $\inv_{S(c,r)}$ and $\inv_{S(c,r)}$ is its own inverse.

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A circle with infinite radius is a line

I am curious about the following diagram: The image implies a circle of infinite radius is a line. Intuitively, I understand this, but I was wondering whether this problem could be stated and proven formally? Under what definition of 'circle' and…
rowe
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Finding the circles passing through two points and touching a circle

Given two points and a circle, construct a/the circle through the two points and touching the given circle. I came across this problem in History of Numerical Analysis by H. Goldstein. I spent some time on this. I have a method of constructing it…
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Equation of the complex locus: $|z-1|=2|z +1|$

This question requires finding the Cartesian equation for the locus: $|z-1| = 2|z+1|$ that is, where the modulus of $z -1$ is twice the modulus of $z+1$ I've solved this problem algebraically (by letting $z=x+iy$) as follows: $\sqrt{(x-1)^2 +…
astiara
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Task "Inversion" (geometry with many circles)

Incircle $\omega$ of triangle $ABC$ with center in point $I$ touches $AB, BC, CA$ in points $C_{1}, A_{1}, B_{1}$. Сircumcircle of triangle $AB_{1}C_{1}$ intersects second time circumcircle of $ABC$ in point $K$. Point $M$ is midpoint of $BC$, $L$…
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If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear

If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear. I really can't seem to get anywhere on this problem, but all I've deduced is that there might be some…
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inverting a cone to a torus

I'm looking at "A Geometric Paradox" by B. H. Brown, in the May--June 1923 issue of The American Mathematical Monthly, pages 193--195. I think people studied advanced Euclidean geometry a lot more then than they do now. The author writes as if for…
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Draw three congruent circles all touching one another, and a second set of three such circles, each touching also two of the first set.

This corresponds to a Steiner's Porism configuration with n = 4, however the trouble I'm having is that while it is easy to construct an n = 4 Steiner's Porism configuration (see second image below), I don't know what the circle of inversion would…
user77970
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Constructing a circle through a given point, tangent to a given line, and tangent to a given circle

While browsing around about problems similar to the problem of Apollonius, I have found references to constructions of all types of circles. For example, not only is it possible to construct a circle tangent to three given circles, but one can…
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Constructing the circle inversion inverse of a point with ruler only

I've been reading a bit about inversive geometry, particularly circle inversion. The following is a problem from Hartshorne's classical geometry, which I've been struggling with on and off for a few days. I figured it would be helpful to show that…
yunone
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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 $ABYZ,BCXY,CDWX,DAZW$ likewise, then $XYZW$ lie on a…
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Convergence of Mixtilinear Triangles to a Point

First, some definitions: A mixtilinear incircle of a triangle is a circle that is tangent to two sides of the triangle and internally tangent to that triangle's circumcircle. There are three mixtilinear incircles for any nondegenerate triangle. The…
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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?
theonlygusti
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How to prove collinearity of circumcenters

Let $A_{1}A_{2}A_{3}$ be a non-isosceles triangle with incenter I. Let $C_{i}$ , $i = 1, 2, 3$, be the smaller circle through $I$ tangent to $A_{i}A_{i+1}$ and $A_{i}A_{i+2}$ (the addition of indices being mod 3). Let $B_{i}$ , $i = 1, 2, 3$, be the…
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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. The Möbius transformation can be turned into inversion…
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What is inversion and how does it act on figure inscribed in a circle?

Trying to wrap my head around inversions. I understand it takes things from inside to outside, such that $\text{distance from some point inside circle} + \text{ distance to new point}=r^2$ Where $r$ is radius of circle of inversion. How would…
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