Questions related to Inversive Geometry and its applications.

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0answers
26 views

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 ...
6
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2answers
131 views

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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0answers
14 views

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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2answers
37 views

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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1answer
11 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 ...
4
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1answer
95 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
51 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 ...
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1answer
59 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
84 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
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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?.
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2answers
90 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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2answers
183 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
559 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
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1answer
246 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
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2answers
959 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
460 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 ...
14
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
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1answer
139 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
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1answer
154 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
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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
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2answers
570 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
597 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
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 ...
3
votes
3answers
706 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
2k 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
302 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, ...