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 $b$, ...
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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 ...
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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.
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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 ...
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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 ...
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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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How to correctly determine the mobius transformation that maps three distinct points to 0,1,$\infty$

I was trying to find the mobius transformation that maps the point $i,-i,\infty$ to $0,1,\infty$. following the formula. $M(z) = K \frac{z-i}{z+i}$. $M(i) = 1 \implies 1 = K\frac{i-i}{i+i} \implies \...
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Inversion in circle

Let $C$ be a circle with the middle point $O$ and the radius $r$, we say that the points $P$ and $P'$ are inverse points with respect to $C$ if: $1.$ $|OP|·|OP'|=r^2$ $2.$ $P$ and $P'$ are on the ...
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inverse point with respect to circle

I'm reading Silverman's Complex variable with application. at page 78, the author says "We say that $s$ and $s^*$ are the inverse points with respect to circle in $\mathbb{C}$ if every line or circle ...
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Circular inversion notes or books

I am preparing a small lesson that I have to do about geometry. The topic of the lesson is circular inversion. Can you suggest me some sources where I can find some nice material about it? Thanks!!
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What is the radius and center of the image of $|z|=1$ under $ f(z) = \frac{3z+2}{4z+3}$?

I would like to compute the image of the circle $|z|=1$ about the fractional linear transformation: $$ f(z) = \frac{3z+2}{4z+3} $$ In particular, I'd like to compute the new center and radius. The ...
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Circle whose radius is infinite

I have the intuition that a circle whose radius is infinite is a straight line. Nonetheless, I don’t feel that what I’ve just stated is really scientific as it has some vagueness and lacks precision. ...
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Inversion in a sphere preserves circles (proof)

Inversion in the unit sphere, for a vector $x$, is defined by $$\frac1x = \frac x{x^2} = \Big(\frac1{x\cdot x}\Big)x$$ How can we prove that a circle's inversion in the sphere is also a circle? (I ...
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What can we do with inversive geometry?

I've recently been introduced to inversive geometry. This seems like it would be a very pretty area of study. Many sources that I have found seem a little old, however. I have two related questions: ...
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How to prove that the inversion $x \mapsto \frac x {\| x \|^2}$ in $\Bbb R^n$ sends circles to generalised circles?

Does anyone know how to prove (or have a reference for) the fact that the inversion $$I : \Bbb R^n \setminus \{0\} \rightarrow \Bbb R^n \setminus \{0\}\\ x \mapsto \frac x {\|x\|^2}$$ sends circles ...
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What does "Inversion about $C$" mean?

Let $\Delta ABC$ be a triangle with orthocenter $H$ and altitudes $AD, BE, CF$. Perform an inversion around $C$ with radius $\sqrt{CH \cdot CF}$. Where do the six points each go? I'm having trouble ...
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7 votes
4 answers
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Proof of Miquel's six circle theorem

Theorem Miquel's six circle theorem states that if in the following all cocircularities except the last one are satisfied, then the last one is implied. In words: if $ABCD$ lie on a circle, and $...
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Circle Inversion

I don't understand the concept of circle inversion. $OP \cdot OP' = k^2$ For example, in a circle $x^2+y^2=k^2$. If I set a general point $P(x,y)$, why is its image $P'(\frac{xk^2}{x^2+y^2}, \frac{...
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Drawing a Poincelet Porism

How to draw a Poincelet porism? In the above image, for the first case, we can draw the inner circle, then the triangle and then it's circumcircle. But how to draw for the other cases and in general?
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Converse inversion

I was puzzling with circle inversions and was wondering about a converse of this: Given 3 points A, B and M Is there only one point R so that $d(R,A) * d(R,B) = d(R,M)^2$ or are there more of them? ...
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equation of inversion of a line [closed]

Say we have a line $L: ax+by+c=0 $ in the plane. Define the inversion of a point $(x,y)$ as $$\frac{(x,y)}{x^2+y^2}= (x',y')$$ with $$(x,y) \cdot (x',y')= 1.$$ What is the equation for the inversion ...
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(Logic behind) Geometric construction for imaginary roots of quadratic.

In the question on this site Geometric derivation of the quadratic equation there are geometric constructions for real roots of a quadratic. As a follow-up I found the construction below for ...
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Parallel lines as generalised circles in inversive geometry

Consider the inversion of circles ($w_1$ and $w_2$) w.r.t. the circle ($c$) that gives circles of infinite radius ($t_1$ and $t_2$ respectively). What is the nature of the generalised circles $t_1$ ...
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Concentric circles in inversive-geometry

Is it possible to obtain two concentric generalised circles(parallel straight lines) through inversion of two circles w.r.t a third circle?
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Second point of intersection of two circles in Möbius geometry

Suppose I have two circles \begin{align*} (x-x_1)^2+(y-y_1)^2 &= r_1^2 \\ (x-x_2)^2+(y-y_2)^2 &= r_2^2 \end{align*} and I also have one point $$p=(x_p,y_p)$$ which is known to lie on both ...
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Find the image of vertical line $x=1$ under Mobius transformation of $\frac{2z}{z+1}$.

Let $f(z) = \dfrac{2z}{z+1}$ and $L_{3}$ = the line $x=1$. I am required to find the image of $L_{3}$ under $f(z)$. I have done this by choosing the three points $1, 1+i,\infty$ and seeing what they ...
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Finding the image of the x axis under a Mobius tranfsformation

Let $f(z) = \frac{2z}{z+1}$ and let $L_{1}$ be the $x$ axis. I am told to show that $f(z)$ maps $L_{1}$ to itself. I have picked three points to determine it image, $0,\ 1$ and $\infty$. I'm ...
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Parallel lines in inversive geometry

Please give the definition of parallel lines(generalised circles) in inversive geometry? Is there a unique definition (because i am confused about the duality of line with circle)? If parallel is ...
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2 answers
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Finding the images under a Mobius transformation

Let $L1$ be the $x$-axis, let $L2$ be the $y$-axis and let $L3$ be the vertical line $x = 1$. For each $k ∈ Z$ let $C_{k}$ denote the circle of radius $r = 1/2$ with centre $z = 1/2 + ki$. Let $f(z) =...
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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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2 votes
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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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3 votes
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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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2 votes
2 answers
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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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