# Questions tagged [inversive-geometry]

Questions related to Inversive Geometry and its applications.

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### Analogues of circle inversion for general conics

I saw a picture of a small object near the edge of a circle along with its circle inversion, and it looked a lot like a reflection. That’s when I remembered that they’re both anti conformal ...
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### Prove that line is orthogonal using inversion

P' is the inverse of P with respect to the circle c and M is a point of a circle c. Line through M and P intersects with c at A and line through M and P' intersects with c at B. Prove that AB is ...
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### Prove that the tangents at $X$ and $Y$ meet on the line $AB$ - circle inversion

Let $P$ be a point outside a circle and let the tangents from $P$ touch the circle at $A$ and $B$. A line through $P$ intersects the circle in points $X$ and $Y$. Prove that the tangents at $X$ and $Y$...
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### My doubt on Inversive Geometry with cyclic quadrilaterals

Let A , B , C be 3 points on a circle $\omega$ . let the tangents at A and at B meet at D . Let DC meets the circle again at E . Then prove that the line AE bisects BD iff AB = BC . I have done at lot ...
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### Circle of Apollonius modified to inversion form

Let $A$ be any point outside the circle $\omega$, let $A^*$ be its inverse around $\omega$, and $P$ is any variable point on $\omega$. Prove that the ratio $PA$/$PA^*$ is constant, and hence find its ...
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### Given intersecting circles $a$ and $b$, find circle $c$ such that $a$ and $b$ are inverses of each other in $c$

(Just to be clear I'm talking about "inversion" as in inversive geometry) Given two intersecting circles $a$ and $b$, how do you find circle $c$ such that $a$ inverted about $c$ yields $b$, ...
1 vote
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### The image of a line through an inversion

I am going to start by saying that geometry is not my strong suit, but I am taking a course on analytic geometry where I learnt about inversions and there is this question that bugs me. The following ...
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### Convergence of Mixtilinear Triangles to a Point

First, some definitions: A mixtilinear incircle of a triangle is a circle that is tangent to two sides of the triangle and internally tangent to that triangle's circumcircle. There are three ...
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### expressing class of inversions in einstein summation convention

I'm working on a personal project where I'm studying the set of inversions in $\Bbb R^n$ that preserve the unit sphere centered at the origin; these transformations can be defined based on a center of ...
310 views

### How to prove collinearity of circumcenters

Let $A_{1}A_{2}A_{3}$ be a non-isosceles triangle with incenter I. Let $C_{i}$ , $i = 1, 2, 3$, be the smaller circle through $I$ tangent to $A_{i}A_{i+1}$ and $A_{i}A_{i+2}$ (the addition of indices ...
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### Which property of polar has been applied to this proof.

Which property of polar has been used please give its proof also.
1 vote
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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### 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?
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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 ...
388 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 $O$...
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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 ...
419 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 ...
184 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 ...
1 vote