Questions related to Inversive Geometry and its applications.

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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 ...
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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 ...
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Circular Inversion

I'm currently working through Coxeter and Greitzer's Geometry Revisited, and came across a problem which would have been trivialised had a certain claim been true. The claim was: circular inversion ...
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Image of a locus via stereographic projections

Yesterday evening I was playing around in my head with stereographic projections and I've come up with this idea. Let $\gamma(t)=(x(t),y(t))$ be a certain curve on a plane. Define a new curve ...
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1answer
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Finding the polar line of the intersection of a polar line and a tangent

Let $K$ be an inversion circle with center $O$ and let $C$ be the point of intersection of two lines tangent to $K$ in $A$ and $B$. Then let $E$ be the intersection of the line $AB$ and the line ...
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Inverse with respect to a given circle

Determine the inverse with respect to a given circle $g:\mathbb{R}^{2} \to \mathbb{R}^{+}, g(x,y)=x^{2}+y^{2}$. I have looked around for non geometric derivations without finding any of value. ...
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Given the circumcircle, the 9-point circle, and the angular measures for a triangle, construct the triangle?

This is similar to some questions that have been asked (e.g. construct-triangle-given-inradius-and-circumradius), but I don't see the exact same question. It arose out of an inversive geometry formula ...
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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), ...
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Is this usage of the term 'Jacobian' related to the other uses e.g. matrix or elliptic function?

In the inversive geometry exercise below (from Geometry Revisited by Coxeter and Greitzer), the solution refers to the point pair $(L, O)$ as the Jacobian of the two point pairs $AC$ and $BD$. I was ...
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Why is inversion in a straight line the same as reflection?

In the inversive plane points $P$ and $P'$ are defined to be inverses with respect to a circle $\omega$ of radius $k$ and center $O$ when they are distinct from $O$, on the same ray emanating from ...
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Hyperbolic inversions are transitive on unit vectors at $x \in D$

Consider the Poincaré model in which the hyperbolic plane is the interior of a disk $D$, and a point $x$ in it with two vectors $v$ and $w$ of the same length attached. The reflection with respect to ...
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Find the circle of inversion that inverts one given triangle into another given triangle.

Given triangles ABC and DEF, find the center O and radius k of the circle of inversion such that the inverses A', B', C' of A, B, C form a triangle congruent to $\Delta DEF.$ (This is problem 5.3.6 ...
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2answers
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Inverting a triangle about a circle centered at the orthocenter.

If the vertices of triangle $\Delta ABC$ are inverted about a circle $\omega$ centered at the orthocenter $H$ of $\Delta ABC$, the new triangle $\Delta A'B'C'$, is similar to the orthic triangle of ...
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1answer
63 views

Differentiation of inverse functions using graphs with conditions?

I was trying to differentiate this equation. And I got the answer but it matches none. Any help on how to solve this one. I tried by converting this function to $y=tan^{-1}tan{\frac{x}2} $ and then ...
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4answers
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How to solve these?

Inverse Trigonometric Functions They are incomplete and I don't know how to complete them. Who can help me? 1st $$ \int\frac 1{ x \sqrt{x^{6} - 4}}dx $$ I tried with: $$u = x^3 $$ $$du= 3x^2dx$$ ...
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1answer
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value of this inverse trigonometric expression.

How to evaluate this expression. $$\sec^2(\tan^{-1} 2)+\csc^2(cot^{-1}(3))$$ I'm stuck on how to process squares, which is on sec and cosec function?.
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101 views

Motivation for the definition of the projective line over a ring

Given a ring (with identity) R, the definition for the projective line over a ring I have is as follows: On $R \times R$ define an equivalence relation as $(u, v) \sim (a, b)$ if and only if there is ...
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A geometry problem uses inversion

Let a triangle $ABC$. $M$ is a point inside triangle. construct the line through M perendicular to $MA, MB, MC$ and intersect $BC, CA, AB $ at $A_0,B_0,C_0$ respectively. Prove that $A_0,B_0,C_0$ are ...
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4answers
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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 ...
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1answer
264 views

Construction of touching circle

My question is: Consider five collinear points $D$, $A$, $C$, $B$ and $E$ such that $DA=AC=a$ , $CB=BE=b$. Let M be the midpoint of $DE$. Let $S_1$ be a circle with center $A$ and radius $a$, Let ...
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construct inverse point with respect to the circle by the use of the compass alone

If the given point P lies inside a circle C ,with center O,the circle of radius OP about P intersects C in two points. How to construct point P' inverse to point P with respect to the circle C by ...
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Is it possible to use inversion to solve this USAMO problem in 2007?

I've no previous experience to solve any problems by inversive geometry but I am willing to see how it works. But I think I know some of the basic definition about inversion in geometry. Also I expect ...
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6answers
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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 ...
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Inversion map in higher dimensions (preserving angles' size)

Consider the inversion with respect to the sphere $S^n \subset R^{n+1}$, that is the map $$ \rho \colon x = (x_1,\dots x_{n+1}) \in R^{n+1}-\mathrm{O} \mapsto \frac{x}{\left \| x \right \|^2} \in ...
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1answer
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Möbius transformation from the complement of the closure of a disc to the unit disc

Let $D$ be the unit disc. Let $U$ be a disc of radius $R$ centered at some $\alpha \in \mathbb{C}$. I want to show that there is a Möbius transformation from $\mathbb{C_{\infty}}$\ $\bar{U}$ into $D$. ...
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3answers
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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 ...
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2answers
598 views

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 ...
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2answers
624 views

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 ...
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1answer
97 views

Is there any invariance under the inversion mapping?

In geometry, there is a transformation called the inversion mapping which maps nonorthogonal circles into nonorthogonal lines and vice versa.(If I make a mistake, inform me, since I am not very ...
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3answers
773 views

Conformality of Inversion Map

I am trying to show that elements of the general Möbius group generated by an affine transformation $f(z) = az+b$, the inversion map $f(z)=\frac{1}{z}$ and complex conjugation $f(z)=\overline{z}$, ...
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4answers
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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 ...
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2answers
324 views

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, ...