# Questions tagged [inversive-geometry]

Questions related to Inversive Geometry and its applications.

82 questions
Filter by
Sorted by
Tagged with
38 views

### Which property of polar has been applied to this proof.

Which property of polar has been used please give its proof also.
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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?
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 $M(z) = \frac{(-4i+6)(z-(1+2i))}{(3-7i)(z-(10-20i)}$ ...
45 views

142 views

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?
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 ...
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 $O$...
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 ...
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 ...
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 ...
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 ...
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://...
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 ...
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}} \...