Questions related to Inversive Geometry and its applications.

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0
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1answer
8 views

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 ...
2
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0answers
29 views

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 ...
0
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1answer
43 views

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 ...
1
vote
1answer
57 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 ...
0
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4answers
83 views

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: $$u = x^3 $$ $$du= 3x^2dx$$ ...
2
votes
1answer
15 views

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?.
0
votes
2answers
81 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 ...
5
votes
2answers
178 views

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 ...
8
votes
3answers
522 views

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 ...
1
vote
1answer
224 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 ...
3
votes
2answers
842 views

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 ...
2
votes
0answers
435 views

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 ...
13
votes
5answers
10k views

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 ...
1
vote
1answer
133 views

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 ...
1
vote
1answer
148 views

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 $D$. ...
6
votes
3answers
3k views

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 ...
6
votes
2answers
544 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 ...
13
votes
2answers
587 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 ...
2
votes
1answer
94 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 ...
3
votes
3answers
661 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 $f(z)=\overline{z}$, ...
11
votes
4answers
1k views

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 ...
8
votes
2answers
293 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, ...