# Questions tagged [inversive-geometry]

Questions related to Inversive Geometry and its applications.

74 questions
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### 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 $M(z) = \frac{(-4i+6)(z-(1+2i))}{(3-7i)(z-(10-20i)}$ ...
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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?
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### 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 ...
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### 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 $O$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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://...
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### 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 ...
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### 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}} \...
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### Prove special case of Brianchon's theorem using inversion

Brianchon's theorem says: When a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. From interactive demo: (...
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